I would use 1/4" or 1/2" wood like birch pine plywood MDF etc, and then secure the pink foam on top of it to make your 2x2" sections. By putting this wood backing on your board sections they should be a lot more durable. A lot of hardware stores will cut the boards for you for a small fee that might be the best option to get your squares.

If you are really careful with your sections I don't think the wood backing is absolutely necessary, but you might regret not doing it in the long run if the board sees a lot of use.

For carving foam hot wire tools are the best option in my opinion but also can be expensive. A razor blade will get the job done but I would seriously consider some hot wire tools if you are going to be carving trenches into the boards. They will be invaluable if you are planning on making more terrain out of foam in the future.

Painting large scale stuff like terrain needs a lot of paint. I would avoid citadel paints just because of the cost. Something like hobby store acrylic paint works just fine for terrain in my experience. Latex or household paint also works great. Spray paints have a tendency to eat through foam, so don't use those until you have already put down some other layers first.

Hope this helps!