Yes, a trumpet mouthpiece is slightly larger than a cornet mouthpiece and will not fit into a cornet. I learned that the hard and expensive way.
7C, 20B, these numbers and letters are all reference to the size and depth of the mouthpiece. Some are shallower or wider or deeper or narrower etc. The C system (7C) is the most common but some brands (like yamaha for example) use different numbering systems so be wary.
The type of mouthpiece really depends on his needs. For example, if he's in high school and been playing for 3 or more years you can probably give him something smaller than a 7C. A popular "next level" mouthpiece would be the 3C Bach mouthpiece. Its good for playing solid notes within the staff as well as a few notes above the staff. But the 20B mouthpiece could be the equivalent to a 3C Bach, so you can always go with a 1C and see if he likes to play that. If you want a good marching mouthpiece Frank Holten 3C or 7C are good. If he just wants to hit really high notes you can look into Schilke mouthpieces. The selection is really endless but I hope this helps a little.