It all depends on the market.

My wife and I have jobs and our credit score is roughly 580.

I don't believe there is a common practice, its going to be dependent on local standards.

Either stay put since you just pulled the HELOC on it or see if you can score a two-family and house hack it and then rent the first one.

Don’t let them pull your credit at each place, which will hurt your score.  

It depends partially on your local PHA, but mostly on whether she is currently on a Section 8 voucher (and her rent has just stagnated) or has she just received one she is intending to use on her current unit?

Depending on what campus you are near will make a huge difference and even more so, how close.

Lots of brokers out there that will do 30 year fixed, but a lot depends on your situation.I can't recommend any bank for DSCR, especially if they're keeping you on a 25 year amortization. 

It depends on which sale you are buying at.

I believe in many circumstances a permit or use license will be required by the municipality depending on the use.