I would like to get into seller financing and subject to deals but don't really fully understand the payments portion of it.

They are subject to a credit, criminal, and eviction check just like other perspective tenant candidates.One thing I would suggest is asking them how much their subsidy is for.  

Because of this, I’m looking for areas close to New York where I can start investing.Since this is my first time, I’d like to focus on neighborhoods I can visit frequently and see for myself.

I do this frequently in Philadelphia where I am located.

In other words, did the appraiser not include any relevant comparables and/or data/facts, that would be better support for the Subject in the market.

Seek out some of the top active Realtors in the Subject's market area and ask them.

If you look through some of the posts on BP, you'll read many stories, specifically of investors from California and NYC buying cheap properties, frequently in the Midwest and South who have lost money.

More frequent turn over, longer turn around, tenants are harder on the properties, and his appreciation is much lower year over year than his NYC rental.  

Learn how to wholesale properties, take homes subject-to, use your license to help others, etc.Good Investing...

The deposit belongs to the tenant, not you, so it shouldn't be subject to any terms outlined in the PM Agreement.If you have a written contract, I recommend you dig through it carefully to understand what you agreed to.