can i use 3 lap distance or is there any minimum for the race distance

3 laps is fine as that is the minimum distance you can select for races.

The rules state you have to do it from the "Championship" Menu. The lowest is 5 laps there, you can only do 3 laps in Grand Prix Mode. Which mode should we choose then for all tracks? Same as any%?

Edit: Just saw its in the Championship category, so it should be 5 Laps.