-Gravitational anomalies is a fair criticism, though this a problem that crops up more time than you may suspect in Star Wars, the Death Star in particular has ISSUES especially when you factor in both old and new canon cutaways.
Inside the structure, I'm not worried. I just always applied that there is some tech method of generating artificial gravity built into the decks of ships space stations. But on the surface of the planet....that's a different story.
-THEORETICALLY if the planet was totally geologically dead (which very well could be), you could go quite a ways before hitting a mantle or core. This would also all depend on how deep the trench is, what precautions were taken, how big Ilum is, how geologically dead it is, how large the core is in relation to the size of the planet, is the core spinning, what its made out of etc etc etc.
Just look at it. They've pretty much removed half the diameter of the equatorial region. Plus all the machinery that is beneath the visible surface.

For these points my advice would be the same with analyzing the travesty of a realistic Death Star: Space Magic
Not sure what you mean by this.
As for the trees, best guess it *is* a whole planet. Star Wars has a bad habit of saying "This whole planet is a Desert/City/Tundra/Ocean" etc. So two possibilities here:
1) Ilum has a more temperate climate than wherever the Jedi Temple was located.
True. I've had similar thoughts. The shortcomings of filming on planet with climate variations.

2) [My personal opinion] The First Order's construction of a mega-base complete with massive Star-Destroying weapon, countless surface emplacements, multiple points of industry and DECADES of industrialism has warmed Ilum's atmosphere enough where life is more practical on the surface.
Uh....no