piano part spaced too far apart

Some of those rests are in funny places. Right-click on a rest, choose Select >All similar elements and press [Ctrl] R to reset them to default positions.

This was created in a pretty ancient version of MuseScore, so it's tough to guarantee good results. But the problem here seem to be that MuseScore is trying to avoid collisions between the lyrics and the measure numbers on the bottom staff. Those measure numbers shouldn't really be there. Simplest solution is to turn off the "All staves" option in Format / Style / Measure numbers - they don't really serve any useful purpose on any staff but the top. But if you do need to enable that option for some special reason, you should then hide them on the bottom staff of the piano part. Right click one, Select / Similar on this staff, "V".