If I have already had a speedrun submitted and it's been verified, and I run a new record, do I just make a new post with that new record run, or do I have to somehow take down the old one? And if I do make a new post, will the former record get taken down automatically or would 2 of my scores appear up on the leaderboard?

Just submit the new run. The old run will automatically turn obsolete once your new run has been verified

It depends on if the board is set up to obsolete past times, but I'm assuming this is the case. You can always submit better runs and have no consequences on the site end.