The only real reason to keep it in your name and do the 1031 is that it gives you more flexibility for borrowing.

I do not allow tenants to move in with borrowed money.

ARV on my most 2 recent buildings were 990K and 1.4million so I am usually borrowing 50% ARV. 

You really should pay that off first before borrowing more money for another house.

Instead, she put up half the down payment and I am the only borrower on the loan and on the title.She paid half the down payment and half of all the mortgage payments until now and the intention was always that this was her house too.

there are some positives, but many negatives to them, you should get a ratified deed up front, be aware that the lender could call the loan, be sure the seller will fully cooperate when it is time to formally transfer, seller could circumvent you at the wrong time, borrow against the potential equity or just say "forget it, i have a cash buyer"!.

I rent to college student here in NY State and an arguement has ensued over a desk between two tenants in which one said they owned it and let the other borrow temporarily and the other is claiming it was given to them.

Or as the Bible says, "The borrower is a slave to the lender." 

Its not hard to find deals that make more than 3.5% per year so if you are borrowing money at that rate, I highly recommend taking all you can out and put it into other deals. 

FHA defines it as "A Principal Residence refers to a dwelling where the Borrower maintains or will maintain their permanent place of abode, and which the Borrower typically occupies or will occupy for the majority of the calendar year."