The mesh is just fiber matting that hasn't totally clung to the shell, I would, as a precaution throw a smooth layer of bondo over there, especially near the edges as the bucket can chip without the full backing there.
Your bucket needs washing because no doubt the helmet has been cut to the tideline (the neck of the plug) and will have had some preliminary dust and gunk on it, it's a precaution to ensure that the paint sticks to the bucket.
Apart from that, be patient, wait a full day before going to the next layer, I would also use heat on the bucket to seal the paint if it is a car spray paint.
Apply primer, rub back with wet and dry, then spray again, that should give you a lovely surface to start your painting.
Finally, it's your choice to cut out the keyslots, but, as Kel says, many do, just to help airflow.