This is fun
Haven't seen this perspective for a while. My old mentor always used to chuckle and say "puppet magic!"
The mikes are taped or strapped to headgear because puppetry if they were on a headset (like the ones in RENT,) you'd hear all the heavy breathing that puppeteers try so hard not to do.
Muppets are some of the few puppets that still use live voice; most puppeteers work from a tape. Personally, I think the ability to ad lib and add life to a character is the whole REASON to use a puppet, but apparently Disney Channel doesn't agree.
Also, I've always worked live, but none of the Henson productions are done that way. I realized the huge difference this makes while watching the Muppet Show episode where Gonzo Evil Kanevils himself into Statler & Waldorf's booth. In my head, I was impressed by how fast the puppeteer raced up the stairs to do the "What a rush" bit. Of course, it doesn't happen that way - they just cut to a different set.
A lot of what you're describing happens with several versions of the same puppet, and a lot of strategic cuts as described above. For instance, sometimes Big Bird has use of both hands on the Street (or Bear in the house,) which can't be bluescreened because of the set. They use a closeup at those times, so they can hide the additional puppeteer.
Forgot - if you get the DVD of The Dark Crystal, you can see behind-the-scenes clips that show a lot about how the puppets are manipulated; It covers most types of puppets. It's hysterical to see Frank Oz with the fat witch puppet up over his head. What an amazing guy!