She transfer to Loyola Univ.

And if after, what would be the process of transferring the property from my name into the company name be?

I say wait until at least 2nd or 3rd property before setting up an LLC, and then transfer the deeds.  

In some trusts nobody has the power to sell property in the trust; as per example a remainder charitable trust where the property has already been transferred to a charity after the beneficiary passes. 

I would definitely make the offer subject to the AC unit transferring with the house. 

When you transfer it to a LLC, that triggers that clause.

I need to tranfer that property to another LLC name and going forward it will file taxes under that name.How does the transfer work?

I typically request payment in the form of Cashiers Check or Wire Transfer.

Any suggestions for creative ways to get around putting the entire 20% down or other ways to cut costs in an internal deal? 

You would be surprised how many there are, however, typically, listings that are in one, do transfer over to other ones, as long as that MLS isn't exclusive.