Hi mikste,
For your convenience I gathered links to info on each site's camping facitilies. Hopefully they should tell you what you need to know about each place.
Monument Valley
Zion Canyon
Bryce Canyon
Grand Canyon --Might want to look here, it states for permits to camp in April, you must apply by Dec 1.
Mesa Verde
As far as weather goes, I believe Bryce and Grand Canyon can dip below 5 degrees celsius as a daily low (at least in April), it depends on where you are in the canyons. The other three should be fine for you.