For doors, particularly a front door or back door, use oil-based paint, at least on the exterior.

I agree, it's more costly but vert effective to use an oil base primer.

I'm not too concerned with the white cabinets...They are oil base and pretty easy to clean. 

Bleach will get it, then use an oil based primer before repainting

We have a property for sale and there is an old underground storage tank that was used for heating oil on the property.

There's an underground oil tank.

one thing I learned...that oil based paint takes a while to fully cure.

If you look it up, you will see that linoleum is made of made materials such as solidified linseed oil (linoxyn), pine rosin, ground cork dust, wood flour, and mineral fillers such as calcium carbonate, most commonly on a burlap or canvas backing.

Didn’t specify the material (black mold, oil tank, w/e), but I assume mold since there were water issues with the exterior.

As with Apartment buildings, lead paint, knob & tube wiring, aluminum wiring, flat roofs, asbestos shingles, in ground oil tanks, etc. can all be issues.