In which categories can I use the game's own timer? Which timer should I use in other categories?

all categories use the built in speedrun mode

I turn on the game's own speed run mode and record a speed run, will it be considered valid even if I only use the speed run mode in the game and do not use any other timer?

Yes it will be valid. In fact, you have to use the ingame speedrun mode in order for the run to be valid. Any other timer is optional.