Risk of insulation triggering mold growth?
The advice to remove water before it comes in is sound but if you run a dehumidifier and the rest of the house isn't extremely tight, it seems unlikely that insulating the attic is going to cause a major difference overall.

It really isn't a problem that your land is flat.  The grade required is very sight because the actual drainage will be to a pipe.  If you live on a street with a storm sewer, it will drain to that which will be below your yard regardless of how flat it is.

The biggest source of water is likely your roof, not the neighbors pool.  If you have old clay drainage, it's probably clogged and cracked and mostly non-functional.  And it's probably installed right up against the foundation.  This means water comes from the drain spouts, goes into the clay pipe and then (because it is porous) bleeds out around your foundation because it has no where else to go.

Contemporary drainage is completely different.  It takes the water direct from the downspout to non-porous PVC and takes it away from the house, typically to a storm sewer.  You can also add drains to other areas of the yard where water pools.  All of this may be subject to code.