For something like the Big Boo Painting star, does time start when you enter Big Boo's Haunt or when you enter the Luigi painting?

Most likely when u enter Big Boos Haunt, because the painting is still from that level. It's not its own separate thing outside the overworld(s). It's in a certain stage, so if I were to guess, the timing should start when u enter that certain stage it's from, just like all the other IL's from any of the 15 courses.

I never really thought about it, but had a quick word with people in discord and most seem to think from BBH entry would be best (the fade in timing just like other level stars), so I guess we will go with that :p