Not only is One-time pad proven secure, it is also obvious without any knowledge of math.
Also good luck finding a nonce-reuse resistant algorithm.
The failure modes of reusing a nonce are never good, but some are far worse than others. The one-time pad's failure mode is particularly disastrous; it leaks the XOR of two plaintexts. There are a good number of algorithms that only leak whether the same message was repeated.
Not that I know of. As far as I know the security reductions are unconditional.
Even worse than revealing the XOR of 2 plaintexts is the key recovery enabled by GCM (source: other comments in this thread). And if you're that scared of nonce reuse, just
Nevertheless, it doesn't matter. Avoiding nonce reuse is easy. And if you're really scared, just use a random nonce from a big enough space (192 bits, like XChacha20, are good). That way if you have access to a random source, you can turn any encryption algorithm into a nonce-less algorithm.