Yes, you can have more than three bombers. Additional units are represented by the white and - (if you’re an exceptionally air-based strategist) - red plastic counters. It gets tricky if you want to spread them over more than three territories because you’ll not have enough of the actual bomber pieces… Logic seems to dictate that you should be able to split your force into however many groups you like, so you would probably just have to improvise some more bombers, or trust to memory.
As to your second question, yes planes can stay in a captured sea territory if it contains a carrier - obviously, it would cost a non-combat move to land on an adjacent land area. You will also know that there must be sufficient carrier-space or in-range land territories for all aircraft to land at the outset of battle (unless you are playing some sort of Kamikazee rule variant).