Can I buy a house for my kid and have them make the payments?

More information needed and It depends on how you are buying the home, primary residence or investment property. You can also be a cosigner and/or gift funds to your kids and they purchase as their primary residence. Definitely talk to your lender about options and the best way to accomplish this purchase.

At face value of this question, yes, you can do this, we have done this hundreds of times for clients.

Depends on your loan terms and requirements. You will want to check with them as some loans require it to be your primary residence for a specified period of time.