He offers Taxpayer B and all other tenants $10,000 to vacate their units immediately, although they aren't compelled to accept, as their right to the unit is guaranteed another 6 months by contract.Is the $10,000 taxable income for Taxpayer B?

That said, I’m not a lawyer so you should consult to be sure.Additionally, asking your lender for permission to stay in the unit is likely not the best avenue, I can almost guarantee that request would be denied, even though you don’t appear to be changing the intent or purpose.

Certainly if he personally guarantees the note.

If you personally guaranteed the new loan, then you a probably stuck as guarantor until you refinance it. 

Am i allowed to do this because my mom is paying for the house and not me, she allowed me to Airbnb, but I decided renting is guaranteed because it's college town.

If you receive a warranty deed when you purchase a property, you are guaranteed the title is free and clear and don't need to be concerned with the previous quitclaims either. 

There are some strategies to increase your initial investment to include renovations, but Fannie Mae is cracking down, so there is no guarantee.

For the first couple of years, you will need to personally guarantee to credit lines as mentioned by @Charles Carillo above. 

Besides, most parents don't want to see their child start off life with an eviction, so you are virtually guaranteed to get paid.

It hasn't been a problem for me with community banks as long as I have been able to guarantee the funds.