I'd like to speedrun Chex Quest 3's 1st and 2nd episodes on GZDoom using Chex 3, but I don't know what category to submit under.

I believe if you speedrun the first two episodes with the Chex Quest 3 file using GZDoom, then the categories would specifically be Chex Quest 1 GZ, Chex Quest 2 GZ, and Chex Quest 3. The first two episodes are different with CQ3, compared to the original versions (1996), so you'll have to specify the version that you are using is CQ3.

What does stuff like UV Speed mean then?

Ah right, I forgot to explain that.

UV Speed - Complete the game as fast as possible on the 4th difficulty (Extreme Ooze)

UV Max - Get 100% flemoids and 100% secrets on every level as fast as possible on Extreme Ooze.

NM Speed - Complete the game as fast as possible on the 5th difficulty (Super Slimey)

NM 100 - Complete the game as fast as possible while obtaining 100% secrets on Super Slimey.

Tyson - Complete the game as fast as possible while zorching 100% flemoids, using ONLY the mini zorcher, or melee weapons. Doesn't specify the difficulty, but I'd assume Extreme Ooze is recommended.

No Monsters - Complete the game as fast as possible, but without monsters. Difficulty does not matter, it's all the same.

UV Fast - Complete the game as fast as possible on Extreme Ooze with fast monsters enabled.

UV Respawn - Complete the game as fast as possible on Extreme Ooze with respawning monsters enabled.

Hope this helps!