Can you live together have children (but not get married.)

Yes why not? A lot of people doit now

You can, yes. Be aware that many states no longer recognize common-law marriages, which means that legally you may not be each other's next-of-kin, which may cause legal problems in life. For example, if your partner is in an accident and gets taken to the hospital, the parents may be the ones calling the shots on medical treatment, not you. Etc. I'm not sure how it affects the offspring of the union, and whether there are any legal hoops you will have to jump through there to get the rights that marriage would give you.

Yes, it's a common practice nowadays.