I'm not sure I am really getting what you're saying, but that may be because you aren't sure you're really getting what your daughter is signaling. So let's take her behavior as you describe it one incident at a time and see what we can figure out.
1. "I have an almost 3 year old girl. She is smart, active, loving and outgoing. She also is randomly and unpredictably violent. I mean violent lightly. She never appears angry and I do not think she is being mean at all. BUT she is more than rough. I find it stressful particularly because it seems so unpredictable. I have to watch her very closely."
It makes sense that you would be very stressed out by your daughter's unpredictably rough behavior. You mention later that it may be predictable but not to you (yet.) It is indeed my experience that we CAN, eventually, predict this kind of behavior, even if the signs are subtle. Usually, children are rough like this when they aren't feeling good inside themselves. So if we can figure out when your daughter isn't feeling so good inside, we can predict and avoid the rough stuff. It is possible, of course, that she is rough ALL the time when she relates to others, but I'm betting that is not the case, or you would not be experiencing it as unpredictable.
For now, since you do find it unpredictable, I would suggest that instead of hoping your little girl will be appropriate, for now you need to assume she will be rough. That means that in every interaction with anyone else, you need to be right next to her, to prevent a rough interaction. If you can stay calm and really pay attention, I am betting you will start to read the signs that something is about to happen. I am assuming from your letter that this is usually when she gets overstimulated in social situations, but it might also be when she is frustrated about something. When you see any sign of aggression or spaciness, you'll want to intervene playfully, as I describe below, to remove her from the situation and help her express her upset directly, rather than against another person.
I want to add that while your daughter is acting aggressively, this is not actually violence. It will help you to reframe this. The fact that your daughter never appears angry is probably the most disturbing thing about what you are describing, simply because it is only natural for little humans to get angry when they are frightened or frustrated. So if she is not appearing angry, it makes me wonder if somehow she has gotten the message that anger is not ok. Has someone gotten angry at her? Or are you so unnerved by anger that you have given her the message that anger is not ok for her? I don't know, of course, but your use of the word violent to describe normal toddler behavior of throwing sand, pushing, etc does seem a bit of an over-reaction. I bring this all up because over and over I see a connection between parents' emotional triggers and their child's behavior. So I'd like to suggest that you take a little time to reflect on your own feelings about anger, and your own history with it. Your goal is to get to a point where you see your daughter's anger as natural and a signal that she needs help handling something. Once you are able to do that, you may see a difference in her behavior.
2."Often when we tell her no to very rough things she has done she laughs."
This is completely normal. It is a sign of discomfort and also some fear about your disapproval. Don't worry about this reaction. She is not taking her roughness lightly; she actually feels uncomfortable about it. That's what the laughter signals.
3. "She met a older (6?) girl at the beach they played a bit then DD threw sand at the girl. And then at her mother. ...She met a boy at the beach, after sort of chatting a bit she started to pull roughly on his leg. I said no. Then she started to pull and hold his arm roughly."
These were strangers. Who knows what triggered her roughness, but I am betting she was in some way feeling over-stimulated or threatened. Or maybe she had enough at some point and wanted them to go away, and didn't realize she could simply leave. When she feels this way, she gets rough -- that makes perfect sense.
So the best way to intervene is this: stay close when she meets someone. If you see the slightest sign of roughness (sand throwing), scoop her up, playfully, and give her a chance to giggle out whatever anxiety she is feeling by playing with you. In other words, be silly and get her giggling. "Excuse me, are you intending to THROW that sand? We don't throw SAND, we throw balls! You come here, you sand-thrower, you!" Make it into a rough-housing fun wrestling game that gets her giggling. Once she's let out her anxiety, she'll almost certainly be able to relate more appropriately. If she starts to throw the sand again, you repeat the game. And you may need to tell the other family "I think we need to head back now, nice to meet you!" as you continue playing with your daughter, moving away from the object of her aggression.
What if she resists your attempt at giggles and instead reacts with tears? Hold her while she cries. She needs to get those feelings out. We play when we can, cry when we have to.
What if she resists your attempt at giggles but reacts with aggression toward you? You keep it a game, if possible, to get her giggling. "Wait a minute, are you trying to throw sand at ME, now? We'll see about that!" Have a wrestling match.
And if she still won't giggle about it but insists on being aggressive--or she keeps trying to throw the sand and be aggressive after you've been giggling with her for ten minutes--it's time to switch gears to help her cry. Look her in the eye. "Sweetie, I'm serious now. That's enough. Sand is for playing, but it hurts when you throw it." She will probably burst into tears, just as she does when you take something away from her. Hold her while she cries.
Here's a description about how to do this that I wrote for another mom, that is not exactly your situation but is relevant and would probably be helpful to you:
4. "She pushed her good friend hard the other day. The only thing I could see is that DD was trying to open the fridge and her friend may have been in the way or stopping her??"
We've all seen other humans as obstacles to what we want, and become annoyed at them. The difference is that we don't shove them, hopefully. And if it is someone we care about, we have empathy for them and we don't really see them as an obstacle. So this is within the range of normal, but it does suggest a lack of empathy. That is not so unusual for an almost three year old, but you probably want to start working with her to develop empathy. Here's an article on that:
5."She bit (lightly but still) her good friend on the cheek during a hug (the same day as above). I have been thinking about this one. The whole story when that we were leaving. Her friend went to hug (my DD LOVES to hug). DD didn't want to. I pushed her into hugging (which I never do) They hugged and DD bit. She has never bitten before! I think it may have been upset because I forced her to hug."
Such a great lesson on never pushing our child into contact they don't want. Clearly she was already over-stimulated (this is the same day as the pushing, above, right?) and she was upset about leaving, which is usually a hard transition because it makes the child feel a sense of loss. I think you are exactly right that the bite was because you pushed her to hug and she just felt overwhelmed by all those feelings.
6. I asked her she gave weird answers like " I want to go up high" but much later DH asked and she said because she didn't want to leave. I DO notice a sort of ..."spazyness", wildness when we are leaving or if she get upset or is told not to do somethings.
I think the spaciness is about a "freeze" reaction. When humans are upset, we feel like it's an emergency and we go into fight, flight or freeze. It might well be that your daughter goes into freeze/spaciness when she is upset emotionally. And her answer "I want to go up high" is a 'flight' answer, meaning she wanted to escape, but couldn't because she found herself clenched in an embrace. When we are in close connection to another person we care about, it brings up all our feelings. That's why if we are upset and someone we love hugs us, we often burst into tears. So when her friend hugged her, it brought up all those mixed up feelings of both over-stimulation and loss, and they felt so uncomfortable to her, she lashed out with a bite. That is totally understandable.
This is a perfect example of something you can train yourself to notice with her. She gets overstimulated in social interactions, and she feels upset when leaving. You can help her with this, and prevent the aggression, by paying attention so that when you are saying goodbye, you move in close to her and help her restore a sense of safety. If she seems unreachable in her spaciness, then if means she needs to giggle or cry to let out that fear, before she can connect with you and feel safe.
The wildness is another form of the same thing. It is a way of fending off anxiety by pushing it into the body, where it makes us hyper. Since anxiety is a mild form of fear, the same approach works as for spaciness. And, in fact, the same approach works for fight or flight, so you'll see it is similar to what we would do in the beach example, above, when your daughter is in "fight" mode.
Mostly, your goal is to help her surface the feelings by giggling or crying. But you can also help her by talking about the feelings and giving them names: "It's sad to say goodbye" ....."You are so hyper, your body is very excited"....."..."You've had enough, so you're throwing sand, but we can just say 'ByeBye!' and leave, we don't have to throw sand at them."
7. She has also pushed over some sitting babies and hit one on the head. She also will hug strange children but also may poke or push strange adults when they walk by (whether in my arms or walking).
Sounds like your daughter could use some help learning about personal space. Teach her to hold her arms out and turn in a circle, and that is her personal space. Then play games where you ask to enter each other's personal space. When you are with her and there is a stranger, remind her that we never intrude on someone's personal space.
I want to add that your daughter might well be poking or pushing an adult who is walking by precisely BECAUSE they are intruding on her personal space. So giving her language for this might also empower her to carve out her own space verbally instead of aggressively.
As far as babies go, many little ones who have empathy issues dislike babies. My theory is that babies evoke their own sense of vulnerability and therefore make them uncomfortable. So you'll need to stay near if she's around babies, and swoop in playfully to get her giggling if she seems ready to push or hit. You can also play games with stuffed animals
8. She never gets visibly angry. She does get upset sometimes when I take something away etc. Then she cries. This is great. It means she is in touch with her sadness and able to cry about it. Hold her and help her feel safe to cry as much as she can. She may use a pretext to cry about something unrelated, which is a great opportunity to offload her upsets that would otherwise come out in aggression.
9. She also will dump out her glass of water. She has done this purposefully and some times almost as an after thought when walking by. Some of what she does also has this detached sense of experimenting.
This doesn't seem aggressive to me. I imagine you aren't using this as an example of violence but rather of un-explainable behavior, right? But at this age, her job is experimenting, and this is normal behavior. Of course you calmly say "Ooops, water doesn't go on the floor. Do you want to dump some water? Here, let's clean this up and then we can set you up at the sink to dump all the water you want." It is also possible she is doing this to provoke you, in which case it is a signal that she needs to cry. In that case, you would handle it exactly as you would handle the aggressive acts: giggle if possible, cry if necessary.
10. Oh and I was just reading how self control is very important factor and I am afraid DD is lacking it or perhaps she is just not listening to me.
It would be very unusual for an almost three year old to have much self-control. Since you're interested in this topic, here are two posts for you to read that talk in depth about self control and helping kids develop it:
I hope this is helpful. Would you write an update in a couple of months and let us know if anything has changed?