2^32+1 is likely reaching the limits of what an elementary school child is able to do by hand. As for the larger numbers: they do get unwieldy quickly, but I think it might be possible to manually find the factors of them with probabilistic primality testing.

Well, the child will be bored much earlier :-)

For Fermat numbers 2^(2^n) + 1, it is well known that every prime factor should have the form k 2^(n+2) + 1. The sequence (A007117) thus goes like: k = 5 for n = 5 (Euler in 1732, very managable), k = 1071 for n = 6 (Thomas Clausen in 1854) and k = 116503103764643 for n = 7 (only known in 1970 after the modern computer).

Just a typo but 2^(2^5) + 1 is 42946967297, you skipped the fourth digit.