2. That all smaller primes are known can't be verified. Primes of this size are usually found by a number of formulas that find "likely" primes and then verified by brute force (afaik). Because of that fuzzy match to start there might be smaller primes that haven't yet been identified.
For example, there is at least one prime between 2^77,232,915 and 2^77,232,916 that we don’t know (https://en.wikipedia.org/wiki/Bertrand%27s_postulate)
(That’s a very loose bound. I think that, for any N > 2, there are more primes than squares of integers less than it)
That’s significant because (n^m) - 1 is divisible by n - 1.
So, for example, 125320078^35785332 - 1 is divisible by 125320077 and, hence, not a prime.
So, this problem has been solved for all n>2.
n^m + 1 is a more difficult problem. For example, for n=10, we get 2: prime, 11: prime, 101: prime, 1001: composite, 10001: composite, etc (at least the 8,000 following are composite. See https://math.stackexchange.com/questions/2108085/is-there-a-...)