Do heavier objects fall slower than lighter things?

No, hevier things only seem to go faster because of air resistance

if you dropped two identical bottles in a vacum, with one empty and the other full, then both bottles would hit the ground at the same time

No. The gravitational pull on all objects - 9.8m/s/s.

Because there is no mass variable in order to find the rate at which objects fall, this means that mass does not have an affect on the rate at which objects fall.

You also must consider wind resistance but that's a more in depth concept.

You can watch a video on youtube called "Hammer vs. Feather - Physics on moon". It shows how on the moon (Where there is no wind resistance) how a hammer and a feather fall at the same rate