You may want a slightly higher sheen (velvet) in the kitchen and bathroom where you may have more moisture and cleaning. 

Mold is caused by moisture and lack of ventilation so yeah thats the deal killer for me personally.

I just received a copy of the previous inspector's report, and there were some plumbing/electrical issues (that supposedly have since been fixed), high moisture in some areas (so we will need to do a mold test), and a roofing issue on the carport, as well as some wood rot and a few other issues.  

yep in your house. so in the summer time AC bills will increase and the house will smell like TIDE detergent while the dryer is on. if you have the units in a closet or area with a closed door all that moisture can cause mold as well. sorry, you really should vent it!

Then just grading like concrete or something around the back of the house to stop moisture from coming into the basement or affecting the sidewalls. 

The other is they now want a vapor barrier installed in the floor but these are older model trailers “as is” and we haven’t had an issue with mold or moisture.

These items can last for a very, very long time if moisture, humidity, and bugs are kept away.I specialize in pre-1950s houses and have one from the 1910's. 

You will need some space for the second unit, preferably close to the first unit, so that tapping in and re-routing the ductwork is minimal.Alternatively, dedicate the existing HVAC to the larger apartment and install a mini-split system for the other.One of the issues with both solutions is that the existing HVAC will be oversized for just the single unit, which can lead to issues with moisture removal during humid periods, but in my case, this has worked fine for several years now.

That way if the heater rusts out or the pressure relief valve blows, it won't (hopefully) flood the apartment under it.Lastly, install a moisture sensor in that pan that will sound an alarm so everyone knows something is up.

But I'd say if you personally are going to live in the home, I'd definitely lean toward doing it.If you see no indication of moisture issues or anything else that makes you think mold is an issue, then I'd say this is optional.