When I'm looking some of the ceilings are discolored and was wondering how do you handle this.

Keep in mind, granite needs to be cleaned and sealed at least 2 times/ year especially in a rental to avoid water stains and discoloration.

So I was laying in bed looking over the photos in detail a week or more before closing the deal and noticed when I zoomed in, the floor joists weren’t just discolored, they were rotted really bad.

Yours sounds worse since there is color discoloration and actual water.

The only two issues are some discoloring by the legs of the bed and some discoloring by the shower, where there is carpet.I am not planning to charge them for it because they have kept good care of the property.

While I got you here I need to ask....What would cause a shot blasted epoxy floor to discolor with a chilled water hydronic leak.

The finished part of the basement would have evidence (discoloration on the drywall, mold, etc. 

They could then do a physical inspection of the wallboard, floors, window frames, etc. to make sure that moisture and discoloration from mold doesn't remain, and to provide their "professional opinion" on that information.Air testing - along with a well written report - should go a long way towards answering any potential questions that might surface.

Sometimes if its localized damage I can refinish the floor and have some discoloration where we repaired...sometimes we have to put vinyl or carpet over it.

The edges of the duvet cover would discolor fairly quickly, due to people's sweat (even if you wash it in between each guest).