1) We feed her in the crate. She wants to stand half in and half out of the crate to eat. I've been picking up her back half and setting her in while saying "place" (the command we want to use to get her to go to the crate
If you want her to eat in the crate and don't want her half-in-half-out then I would put the food in there with her and close the door until she is used to eating that way.
2) Today she stopped wanting to go in. I want her to get used to napping in it. She whines for a bit when I put her in (5-10mins) but settles down and sleeps. We have her in there at night too...is that okay?
That is okay. As long as she is not in acute distress it is okay to hold firm boundries with the crate training and naps. Use lots of positive reinforcement to get her to go in the crate on her own, and make the crate as pleasant a place as possible for her to be. We fed treats in the crate, gave her treats for going in on her own, kept interesting "crate only" toys and chews that we would rotate in and out of the crate, etc.
3) She doesn't wake us up to take her to potty at night until she's peed a couple of times in the crate. Should we be setting alarms and making her or will she get it?
What size is the crate? For potty training purposes it should be just big enough for her to stand up, turn around, and lie down comfortably in it. Any bigger and a puppy is more likely to potty in her crate. I had a large crate for my pup (she was 8 lbs when I brought her home, and is 70 lbs now) but it had a divider so that the crate could start small and grow with her.
If the crate is the proper size and she is still pottying in her crate at night then setting an alarm every three or four hours and taking her out is a good idea. You really do not want her getting the idea that her crate is the place to potty, and it might take some real work for a couple of months to break the habit. As she gets older and has more control over her bladder you will be able to wake less often, and hopefully she will let you know when she needs to go out.
4) How long until the play biting is done? She nipped my toddler today because he got down on her level and she was in extreme play mode.
You've got a while till the play biting is over! It is so hard with little kids because they really want to play with the puppy, but puppies are made to wrestle and play with their teeth. 100% supervision is key in the stage, as is redirecting the puppy, giving her plenty of toys to bite, chew and mouth during play time.
You should work with you pup on developing a soft mouth. When the puppy play bites you give an abrupt, high pitched "YELP" and quite playing immediately. This is puppy language for "you play too hard!" She will eventually learn to hold her bite (bite inhibition), which is incredibly important. With our pup I trained my daughter to yelp EVERY time she was mouthed. Merlow learned (after a few months of repetition) that mouthing kids is NEVER okay.
For my husband and I we tolerated a certain pressure of mouthing before yelping, and as she got older we responded to lighter and lighter pressure, until finally we permitted no teeth on skin whatsoever. This trained our dog to respond to us with lighter and lighter tooth pressure, and she really knows how to hold her bite. Puppies learn this from playing with each other, but unless you have more than one puppy it falls on the humans to teach the skill.
5) Should I be shutting the door on the crate when she's in it?
If you need her to be in the crate while you are doing something else and cannot directly supervise her, then yes, close the door. My pup had her crate closed during naps, at night, when I was in another room and couldn't take her with me, and when she was being too wild with our daughter and needed to calm down. As she gets much older and is fully house trained (no more pottying inside, no eating shoes and furniture, good people manners) then the door can stay open more often. In the mean time make the crate a comfy place to be: soft, warm, with a good number of interesting treats and chews and toys that you can rotate and keep her happy and interested in.
Have fun with your new pup!