I have just one on speed dial and that's cause she does our rentals. 

So I am looking at a duplex that got water/mold damage on one side, and minor water damage (Door left open or something like that) on the other side.

If so let current move out and charge through the day before the new person moves in plus damages.

Or do we keep the costs down and keep vendor 1 hoping the new unit doesnt cause problems later on?

Mine is to get paid my rent in full at the beginning of the month, have a tenant that doesn't damage my property and keeps my expenses minimal, and avoid vacancies by keeping the tenant happy enough to renew at a higher rate each year.What will you accomplish by enforcing a late fee on the 3rd when the 1st falls on the weekend?

I was a little concerned if they'll accept the offer cause I really liked the property.

I'll still file an eviction as well and go for the second clause(damages).Good Luck. 

If you don't require it there should be a clause in your lease either way regarding damage to their personal items etc.

Was it caused by a leaky roof or window? 

Well, my iPad spellcheck caused one if many errors that I did not catch.While it is true that I do not look for or at the privates of my prospects, that is not the point of my post.I market to attorneys of the people in charge.