No loose boards, popped screws, solid railings.

Am I just screwed here or is there a way I can come out on top with this situation? 

If they put large nails or screws in the wall, then it's considered damage and I charge for it.

It's a few hundred dollars, and probably worth it not to screw it up given the dollar amounts you are dealing with.  

This is why I am looking to propose a reasonable price, I just do not want to screw myself in under estimating rehab price.

This is about the only time in my life where I will be able to save money without finantial burdens and I just dont want to screw it up.

Replacing the vanity itself is very straighforward - find your studs, set the vanity in place and shim and adjust it so that it's level and plumb, and then use cabinet screws into the studs to make it permanent.

Margins are tight and screwing up the math or having unrealistic expectations will end you up in trouble pretty quick!

They should be able to identify what happened, who screwed up, and what your options going forward are.

And lastly, make sure the property management company has experience leasing and managing Section 8 properties, since there is a lot paperwork and processes, and I've seen experienced property management companies who do this stuff all the time, screw up on multiple Section 8 leases to where the tenant's packet expired because they were so incompetent.