I’ll try to answer your questions as best I can:
Your plastic crate should be fine, especially if you block off the back. We’ve always used wire crates, both because they are less expensive and provide better air circulation, but many people have successfully used (and even prefer) the plastic airline kind. Definitely start out with what you already have.
Socializing is always a balance between avoiding unnecessary exposure to germs while exposing the puppy to life experiences. I wouldn’t let the puppy roam around on the ground where there are likely to have been unvaccinated dogs in the area, but you can (for example) take them to Petsmart and keep them in the shopping cart so they can see but not touch. We’ve always walked ours in our neighborhood (never seen any strays and people take care of their animals) but not in the park where who knows who’s been there. Just use common sense, and carry them into the vet - that’s where they are most likely to be exposed.
As far as playing, I would say don’t do anything now that you wouldn’t want them to think is ok as an adult ... so even if it’s cute that they are mouthing your hands now, you probably don’t want to encourage that. Other than that, we let the puppies be the guide - if something is getting them so wound up that they seem to lose control, we back off a bit, but we have always gotten down on the ground and rolled around with our pups. We also do play tug but only with designated toys.
You can start training pretty much anything you like. Their name (and to pay attention to you when they hear it) is definitely a good first step, but at 8 weeks old, mine already knew (we raised them from birth so had a head start) sit, down, spin, twist, come, and touch (with their nose to our hand). They are little sponges at this age ... it’s actually easier to teach them stuff when they are little than when they are older. Just don’t do too much at once, their attention spans are quite limited.
As far as run of the house, the key is whether or not you are supervising them. If you are right there watching (and I mean actively watching, not just in the room doing something else), then it is fine to let them explore, but be ready to grab them and rush them outside if they start to circle, sniff and/or squat. If you aren’t able to supervise, then in the crate. Once they get a handle on the whole house-training thing, then you can gradually expand the area they can roam in, but be sure to puppy proof it well if you won’t be right there on top of them. We use baby gates (open floor plan so few doors on the 1st floor) to mark out proofed areas they can run around in, although we don’t leave them completely unsupervised for quite some time.
Hope this helps ...