@AlexanderCurl, regarding sequential key generation:

Have you tried adding a large prime number, like 0xdeadbeef (just an example, I don't even know if that's prime)?

Sure, you'll have to check for overflow more frequently - fortunately, that's just a matter of doing a Greater-Than comparison followed by a subtraction - but a sufficiently large increment should make the keys look pseudorandom as far as the bits are concerned.