First, I want to say that The Mamas and The Papas are one of my all-time favorite groups, и я обожаю Cass Elliot -- она была богиня, и всё. So it made me very happy to see that they have some fans in Russia!
A few comments about your specific questions:
No, the opposite -- "to be itchy to do something" means that you are very eager and impatient.John and Mitchy were gettin' kind of itchy
Just to leave the folk music behind; (John and Mitchy didn't want to leave folk music? Were they uncomfortable to leave folk music?)
Yes, exactly -- "to pass the hat" means "to request voluntary donations of money."And after every number they'd pass the hat. (what is "pass the hat" - to gather money from people?)
"Love bumps" can be interpreted as (1) a euphemism for a sexually transmitted disease like herpes or genital warts; or (2) more innocently, госиная кожа от любви. I don't know which sense is meant here (but most likely it's the second meaning).When Denny met Cass he gave her love bumps; (love bumps?)
"Mugwumps" was the name of their group.Called John and Zal and that was the Mugwumps. (mugwumps?)
Here, "to catch fire" means "to become popular and successful."McGuinn and McGuire just a-catchin' fire in L.A., (cathcing fire?)