"What is the use case for Path mtu discovery and mss adjust."
Just what their names imply. Sorry, sort of a circular answer, but if you don't understand PMTUD and/or MSS sizing, probably best to search the Internet for explanatory material, as cannot easily explain use cases in a sentence, or even a paragraph, or two.
"Do they do the same thing?"
No.
"When would one need to enable DF bit in ipv4 to avoid fragmentation?"
"need"? Why when one wants to avoid fragmentation. I.e. you don't "need" to avoid fragmentation, but lots of reasons why you want to. Sorry, again, a subject that you should searched for additional material.
"I have never had to do this before, is this something that happens automatically behind-the -scenes?"
Laugh, didn't I just question "need". I.e. since there's no "need" you never "needed" to do anything.
Anyway, things (usually) "work" without getting mucking with PMTUD or Cisco's "TCP adjust-mss", but things often work much better when one or both are (correctly) used.
From your questions, I'm assuming that much of the forgoing TechNote might be difficult for you to understand, for lack of knowledge. (BTW, nothing wrong in lacking knowledge, we all start with a blank slate!) Again, so much may be needed to be understood, you may need to find primer material.
The TechNote @MHM Cisco World recommended is excellent, but if most of the subject material is unfamiliar to you, the TechNote might confuse more than help.
