Drumset Rests

To know for sure we would need to see the actual score (.mscz file with no comma at the end). From looking at my picture I would guess that you (or someone) created the first measure using chords (it looks likes everything is in voice 1) and voices for the second measure.

the 2nd measure has 2 voices (voice 1: hands and voice 2: feet), so separate rests for them, and yes, I believe this is correct, for drum staves.
If you don't want the rests, make the the voice 2 ones invisible (select press V), the voice 1 ones will automagically move down then

My 2 cents: I would always notate the drum bass in voice 2 (in the first measure of your example too), and yes, I would stay visible the rests of voice 2 (maybe in your example it's a case of personal preference) . In your example the bass drum, side stick and hi-hat are on the same beat. But there are other patterns, where it differs. Especially in such cases I think it's easier to read the pattern for the musician.

The general rule is "stems up (voice 1) for notes played by the hands, stems down (voice 2) for notes played by the feet". This is what MuseScore normally does by default, and in such cases, it is usually completely appropriate to show the full set of rests for both voices. However, if the two rhythms happen to be substantially the same, then combining the rests and occasionally even the notes into a single voice is not a bad idea either. Like in your example where both hands and a foot are always hitting at once, I'd consider using the notation shown in the first measure (created, as mentioned, by using chords in a single voice rather than the default two voices).