This kind of thing happens in other genres but since I mostly listen to rock then I’ll use it for an example.
Is there a name for when you have a rock song where the first verse is mellow, you get to the chorus which rocks, the second verse mellows out again, repeat…
I don’t know that there is a specific term for it, it’s just using dynamics to add an extra dimension to the music.
Yeah I think you call that a “power ballad.”
That’s not bad, though it only refers to rock music.
No, not one of those.
For example, “Sh*t Town” from Live.
If you haven’t heard the song and check out the link, listen to the audio in the upper right, and not the video in the lower left.
Here’s a pop song that does this:
“Special” by Wilshire.
I’m not aware of any term for this. It’s just one of many ways to create contrast between verses and choruses in standard pop songwriting.
On second thought, never mind. 
OK, I guess not all common techniques have a name.