Well... my daughter turned 2 in May, but I can answer some of those questions.
Does your 3 year old:
-move the step stool and put the potty seat on the toilet on their own?
She doesn't use a stool. She started climbing on the big toilet by herself when she was about 17 months. Had she needed one, she likely would have moved it herself. She was very into doing the potty-business on her own.
-wipe their bottom by themself (at least with pee)?
Not usually. I have to remind her (if I catch her going to the bathroom) but more often than not, I go behind her and do it again to make sure it's done well. I think many children in the 2-4 range have difficulties performing this task properly. It's a muscle development issue more than a behavior issue.
-get dressed or at least TRY to on their own?
She can dress herself sometimes. She will try on her own if we ask. She can get it right sometimes. Mostly, though, she likes to undress herself. She's like a little naked ninja. I can turn around for 5 seconds and when I turn back, she'll be completed nude. *sigh*
-put shoes on without help or at least TRY to?
She's done this for quite awhile. She just loves shoes, though. She obviously needs help with tennis shoes or shoes that are too snug, but we haven't worn many of those recently. Usually I'll point out the pair of sandals that I want her to wear and she puts them on herself. For some reason she never gets the sides wrong, either. (Which baffles me since I suffer from a form of dyslexia and that task took me forever to master. LOL)
-will get a toy they want from their room on their own or will they beg and plead with you to get it when you are busy dealing with something else?
I can't think of a single time that she's asked me to get her a toy. I can't even think of a reason for her to ask me that. The only things she asks me for are things that she can't reach or get herself. (And that's after exhausting ever attempt to get it on her own.) Every morning when she wakes up she belts out:
I wanna play wif barbies! and she runs into her bedroom to get them herself. (Unfortunately she has been ended most of her nights in OUR room, in spite of starting them in her room... but that's neither here nor there, I suppose.

)
Honestly, Aurora is independent to a fault. There are things she needs help with sometimes, but she really has to try herself to death before she'll accept assistance. When she was teaching herself to use the potty, it had to be completely on HER terms. If I so much as suggested a potty break, she would stubbornly refuse-- even if it meant having an accident, which she didn't enjoy. She taught herself many tasks earlier than what seems to be the norm, simply because she doesn't like relying on help from others. She's quite stubborn.
I think the average toddler/ preschooler is somewhere between Aurora and having to depend on their parents for
everything. Her older sister was very much in the middle. She allowed me to help her with some things, while slowly branching out into doing things for herself. She was never a
"Do this for me, Mommy!" kinda kid either, though. She just wasn't quite as motivated to be independent at such a young age. I know I had to assist her with wiping on up to at least 4. She PLed at an average age between 2-2½, but she just wasn't coordinated enough to efficiently wipe her own butt after pooping. Truth be known, she's 7½ now and she still isn't fantastic at it... but I draw the line at wiping the butt of a child not too far off from needing training bras full-time.

Obviously there is a wide range of "normal". But at 3, I think perhaps your DD should be able to retrieve a toy for herself if she desires it. I think it couldn't hurt to encourage a little more independence. I wouldn't stress about it though. Children all develop emotionally and physically at very different rates.