A car does not normally leak oil (this is the year 2016, not 1951), and in any case oil does not belong on any surface (certainly not where there's foot traffic).

Surfaces were not wiped and there was cat hair everywhere (no pets were allowed.)  

Some of the fibers were blue through and through but a lot of fibers just had the paint on the surface so it's looks a little better despite the carpet also now being shorter in that area.

You know, since that guy is acting like he has the right to go do what he wants on your property, what you might do instead of putting up signs, is send a letter to the adjoining property owners, just saying you've bought the property and want to stop the foot traffic through it, and here's your contact information, and please let them know if anyone goes onto the property, as someone's been vandalizing your plants.  

New flooring, carpet, granite in the kitchen and new solid surface vanity tops in both baths, all new appliances, new paint, new carpet.

It is easily killed, especially if it is on a non-porous surface.

Most mold we've found in a tenant's dwelling has been surface mold caused by something the tenant did or failed to do.

I would make the sister an official resident and deal with any real problems that surface. 

What I do, is try to look at the worst case scenario.And it's my opinion, that in the worst case scenario I can think of - where some tenant says their kid or dog got sick from eating soil on your property, I could argue in court that there would be no reasonable expectation for me to "assume" that the property was unsafe - given the fact that the city gave the okay, and I have the documentation to prove it.I would be quite happy with that risk.You can drive yourself crazy over this one, wondering if even if you test the surface soil and find nothing, that there may be some mystery leak buried down in the soil that may somehow rise to the surface, only to end up in the mouth of some dirt-eating child...  

It's amazing how many people use metal utensils on a nice new nonstick cooking surface....or cut things directly on countertops rather than use a cutting board.