IIRC some work was done to evaluate shielding only the relatively small area around the crew and avionics, saving considerable weight. However, it was soon discovered that radiation exposure can dramatically reduce the fatigue life of metallic materials. So, some (large) amount of shielding is still necessary.
Our computers are even more sensitive to radioactivity than our bodies.
Maybe yes with a drone based on TTL or some other "old" standard. Probably no with CMOS... Well, with a nuclear plant on board, I guess one can generate enough energy for a TTL computer.
But computers don't take up nearly as much space. You would only need to shield that one specific spot where the computer was rather than the whole cockpit.
Yes, every particle crossing the copper can create an anomalous signal that can switch a 0 to 1 or visa versa. If you have enough of those, the program(s) will eventually crash. On the processors themselves the L1/L2 caches are vulnerable, but beyond that, the ROM could also get corrupted making hard resets impossible even after a crash.
Unmanned aircraft were impractical during the Cold War, which is the only point in history when nuclear-powered aircraft were seriously considered. Drones are workable now, but political considerations alone make nuclear aircraft a non-starter today.