It might let them know you are getting more serious about things,  though its pretty likely the way this ends is that sometime after these rules lift they will disappear in the night and leave you a mess to clean up along with the unpaid debt.   

The carpets are my biggest concern due to the horrible smell, do I need to attempt to clean them or can I just go straight to replacement?

@Jake Wong If the house was clean enough to be placed on the market for sale, it should be clean enough for a rental.

The interior is a bit dated but is extremely clean and well cared as it has mostly been vacant outside of when they visit the area for vacation.

At my own home I do early routine cleaning and maintenance which runs about 100$ a year, so wondering if this can be done every 6 months or every year to clean the drain pan, filter, etc. at what frequency? 

@Joshua FerrariHow well do you clean your property before tenant moves in?

If you want to chase down probate leads, you've got some homework to do first, like the probate laws for your area and how someone can navigate the process (your counties website should have some guides for this), how to find good estate sale reps, how to do clean outs for cheap, vendor lists, etc.. all the things someone that is overwhelmed with the role that has been thrust upon them and how to best help them.

I’ve only been advised to make sure the funds are cleanly separated

If the tank, assuming it is oil, was not fully washed out and cleaned, the sand presses into the oil and the oil will break the bottom of the tank and hit the soil below.

It was a big project and above all did a great job and cleaned up afterwards.