As long as you don't get a top 5 time (I believed) you don't need proof. And I would reccomend starting w/ NG+, as it's a very easy category to learn.

Well video proof of your runs are required so you could use a capture card! If you don't have one, you can always just record your TV screen too :D Remember to have fun and stay fresh!

Livesplit is the best way

Or if you don't have that, and you don't play near a computer, MSplit for your phone is a great choice.