You can wash your hair every two days, but aiming for washing it twice a week would be better. Washing it too often will strip your hair if its natural...
I don't think it matters whether you fight Bowser before getting all the stars or not. As long as you have 120 stars when you defeat him, you'll get Luigi.
Yes, you can shower with fake lashes. Just know that you may risk getting a build up of makeup stuck in the lashes. Make sure to clean the lashes thoroughly...