I plan on doing a run with two friends at some point, and Realms is the least laggy of our options. I'm not the owner of the realms so I don't know how to show that its a random seed, or that the world hasn't been tampered with pre-game. If there are any other rules or things I should know about doing a run on realms, that would be much appreciated.

I would recommend seeing if your friend can either record the whole thing, or just the beginning and have it showing at the same time as yours, like some multiplayer runs do, showing the very beginning with proof that it's random. after that you could just cut his out once there's valid proof that it's a random seed, and that is the seed you're playing on.