I don’t think it’s your battery nor the screen
have you tried resetting the PRAM ?
do you have any boot manager or EFI modifications programme install with your administrator account ?
or have you recently dual boot a different OS ?
try all of the above, and if it’s to no avail try pressing and holding down cntr + s when you notice any sign of response after pressing the power button that should try placing your macintosh to default startup behaviour
hope one of these help fix your problems
cheers mate