If I were you I would proceed with either of the following options:

1. Ban the use of mouse keys and other similar programs and delete all sub 4 second runs. All runners must submit a full screen video to make it clearer for mods to verify.

2. Create a new category called "With Mouse Keys" and shift all sub 4 second runs there. Also require all runs to be recorded in full screen.

I was thinking if maybe the mods could add a new category called "no mouse keys" so that people without the program can still earn a good position on the leaderboard.

Yea, I don't think its possible. You may want to search "diversity minecraft 15 minutes" on youtube, where someone got all but 2 wools from survival branch (even though its not valid as he placed blocks in the bedrock chamber).

If it happens to be possible to obtain wool blocks from a place not from the required branch (for example obtaining yellow wool in another branch that is not arena), will it be valid? Just asking so I know whether to spend time searching for such an exploit.

I am intending to speedrun a custom map. Is it ok to use 1.7, which does not affect the map even though the map was made for 1.4 (the map did not specify a version to use)?

Also, does using ender pearls to pass through blocks count as a glitch?