I can help with your chemistry question. Insulation foam comes in two flavors, styrene and urethane. Styrene/polystyrene (Styrofoam, or Owens Corning pink or blue insulation board, commonly carved for large projects like motorcycle seats, prop casts, stuff like that) melts when exposed to pretty much any aromatic hydrocarbon. That means anything you'd find on the back of a can of spraypaint, or sounds like you'd find it there. "Petroleum Distillates" is usually a good indicator that you might find a light aromatic hydrocarbon inside.
Urethane foam is a little more resilient, but not by much. You still want to avoid light hydrocarbons, but urethane foams like Great Stuff or DAP that come in a spray can offer slightly improved resistance against heavier hydrocarbons such as oils.
It boils down to what the solvent is in your silicone.
Many of the silicone products I've used in the past have used acetic acid as their solvent, and trace amounts of platinum as the curing catalyst.
If your silicone smells like vinegar when you open it up, chances are good you're using a similar type, and that it will NOT react adversely to the insulation foam.
If you look up the MSDS for the product and examine it for contents, look for things like butane, heptane, xylene, toluene, methyl ethyl ketones (or any ketone- or acetone-type solvent. or frankly, anything with a methylene base, ethylene bases aren't AS aggressive but may still pit or weaken your foam, I THINK).
Silicone itself is a hydrocarbon, but a largely inert one, so what you're looking for are the chemicals mixed into it to get it to flow or harden quickly.
As far as the cuts go, what I would recommend trying, is making a tool like the ones you see above, with a hard steel wire. Make a loop the size you want, grip the loose ends with a pair of Vise-Grips, and peel away the cut a little bit at a time because if you try to do too much the clay will bend the wire.
If you use a coarser wire you will need to grind or file the leading edge to some kind of angle so that it actually cuts instead of just mooshing. FIner wires may not work so well unless they're very, very stiff.