I am planning on doing LTS level speedruns and I wanted to know if I can do level speedruns on challenge mode? Also, when do we use game time and real time? Thanks

Yes you can and use game time

Okay thanks! Use game time for every kind of run in sonic heroes?

the rules of the category elaborate how timing should be done, but it basically comes down to the below

for every individual level, use the stage timer for story categories, use the file time for all a ranks, use real time for super hard mode, use the in-game time calculator that was created specifically for the category

Okay I see. How does this relate to live split? If I set live split to real time, every time I press the key to stop the split, it stops. But when I use game time, if I press the key it does not stop. Should live split be set to game time or real time?

livesplit only matters for super hard mode, the rest of the categories don't need it

if you want to do splits for comparison anyway (most runners do), use real time