Corey,
Your attorney can handle all the paperwork or draft your documents as needed in your jurisdiction. You send the payments directly to the lender yourself, (to make sure they are being paid) probably electronically. As part of the other info you had signed when taking the home subject 2, you also have authorization to speak on behalf of the original homeowner to the lienholder and let them know you are now managing the property for them and you give them your company info and inform them you are the primary contact regarding questions for the property. I would also get a limited power of attorney signed by the homeowner so you can get the escrow overages or other credits when you eventually sell this home (paying off the mortgage) and so you don't really need to find the seller as you have all the docs and info needed to close without talking to them again, if all goes well. The POA is more critical in a lease option as opposed to this deal here because you already have the deed so they really can't do much to stop the sale (other than sue you of course) but I like to cover all the bases up front. Also, it is very important to be upfront and honest with them about what you are doing. You don't have to scam them to make this a win-win.
Regarding your question about them trusting that you will make the payments, the simple answer is that yes, they just trust you. If someone is willing to deed you the home, they generally don't consider the fact you might not make the payments. They want you to be credible as you are saving them from having a foreclosure on their record. More savvy homeowners will ask about this and you should be prepared with an answer. You will find other threads discussing what some of us say that I won't repeat here. It is a very legitimate question that I would be asking if I were in their shoes so be prepared for it.
As Jon pointed out, you have the deed to the home while the sellers mortgage is still in effect. Another benefit to taking the home subject 2 is now that you have the deed, you can take the depreciation write-off, which you could not if you just leased it or lease optioned it from the seller.
They can't screw you or keep the home; they have deeded the home to you so they no longer have title or any interest in the property, other than the mortgage still liened against the home.
Also as stated, because title has changed ( you got the deed), the lienholder (the holder of the mortgage) has the right but not the obligation to enforce the due on sale clause, which would allow them to accelerate the loan due now or very shortly, forcing you to refinance or bring in new financing. Remember, banks like to make loans, not own property so the chances of them calling a loan due are very remote. Worst case, if they did, you would have some time to work things out with them. Don't let this scare you off a subject 2, as they have no real financial incentive to call a loan due unless mortgage rates increased dramatically, as Jon explains quite clearly.
I believe that answers most of your questions. I am a little slow replying.
Good investing
Mike C