- See https://cs.stanford.edu/~knuth/email.html — just send a letter to his Stanford address.

- You may have misinterpreted the talk. There's already a proof (even for the closed form of the leading coefficient), by someone (I have the name written down somewhere but it's in the video). You might enjoy submitting a solution to the AMM problem instead (which only asks for the rate of growth, but if you have a fuller proof I'm sure that's good).

I once sent him a letter (snail mail) addressed to Donald Knuth, c/o Stanford Computer Science Department (address at bottom of http://www-cs.stanford.edu/). He got back within a couple of weeks, so it seems to have reached him fairly efficiently.