The aerator is the small piece on the underside of the faucet stem that the water flows out of.

It appears that the mold is only confined to the underside of the particle board countertop around the sink edge.  2nd.

Also be sure to check that the underside of the roof is insulated with some sort of condensation barrier.

You can also pick up sound dampening padding on amazon fairly cheap, you can install this in the ceiling on the underside of the floor boards. you just have to get something in there to deaden the sound, otherwise think of the ceiling as a drum and when someone walks on the floor it is chambering the sound and releasing it to the downstairs, same can be said for the TV and radio, it is just injecting that drum with sound and it has to go somewhere.I hope this helps,Nick

Especially during the winter, the hot humid air from the bath is going into the attic hitting the cold underside of the roof and condensing.

If you are able to access the attic, you can see if from the underside and get a better idea of the damage. 

Foam on the underside of the decking would be the best solution from an r-value standpoint, but it's wildly expensive.

Up here full basements in older homes were frequently conditioned, but rarely insulated ... or, often we find the underside of the first floor insulated which is the last place you should insulate in a conditioned basement.

Another approach advocated by Joseph Lstiburek of the Building Science Corporation, is to leave a dead air space at the top of the joist cavity with insulation filling the remainder of the cavity and a 1.5+ layer of hardboard insulation spanning the underside of the joists.

The underside of the pier and beam house had more than a foot of water in places.