You need to get written permission from the lender before you transfer anything. 

At what point in the process can I make the gift / what is the best strategy to expedite the transfer? 

For good tenants,  I keep the rates just slightly below market and they'll stay unless of job transfer. 

Or if title has been transferred to the kids and one of them filed bankruptcy this could become an issue? 

Handle a standard operating agreement, transfer a deed to an LLC, and setup an LLC.

This is someone who is already on Section 8, so they already have a voucher, but it would have to be transfered for her to rent my unit.

So what I'm trying to do is once these properties are his and I have closed on my house I would like to transfer them into my name without having to pay a closing cost with the bank again.

It seems logical that this would make the tax regressive past some breakpoint If the government continues to deflate the currency going forward and the tax on inflation tax becomes more common will we see even more "revenue" transferred to the government coffers from those under said affluence barrier?

It's customary to use a Quit Claim Deed for such transfers, but you could do your title company a favor and use a Warranty Deed, subject to existing financing.

One factor is the state where the sale occurred, whether the sale transfers all the liens to the tax sale buyer, and how good is the notice.