In the future yes, there will be a separate leaderboard for mouse and keyboard. It will be under MISC categories so look forward to it being there!

If you quit from the menu, the in-game time you spent in the level is saved, so it is allowed.
If you restart from the pause menu, your run is invalid in any category as this does not save the time spent in that level.
Although the use case for this is very small; For example, the only place this can really be used is when you cant get a gold or platinum trophy in 100%, and the level will take a very long time to play out, so it would be optimal to quit and take the 2 minute time loss, as opposed to a 4 minute time loss from finishing the level (this can happen if you fall too far behind on Rice Cracker Rush, or any of the duels). Other than that, on story it really only should be used if the level is not possible to beat with your current health, and death is not an option.

Edits:
Changed name of All Challenges to Challenges%
Changed name of All Leagues to Leagues%
Changed name of All Challenges and Leagues to Trophies%
The above categories are no longer MISC
Changed name of Any% to Story%
Changed name of Any% Co-op to Story% Co-op
New Category:
100% ~ Consists of getting the Player profile to 100% completion and has 2 sub-categories:
- Golds or Better
- All Platinum Trophies
Added Sub-Categories:
Challenges%, Leagues%, and Trophies%
- Any Trophy
- Golds or Better
- All Platinum Trophies
All runs in Challenges%, Leagues%, and Trophies% categories before 9/15 were moved to the Any Trophy sub-category.
Thanks to everyone for continuing to speed run the game, and I look forward to seeing some runs in the 100% category!
Any questions? Feel free to ping me in general~ (TS speedrunning discord channel)

- Emo