Personally I'd like Blue Mage, Beastmaster and Puppetmaster to share a hub somewhere. A place where they all hang out and make jokes about the real jobs.

Being able to use limited jobs with squadrons would be good, I don't really see how said change would negatively effect anyone. I'd also like to use them in PotD and HoH.

Not bothered about PvP really, being able to duel at wolves den could be cool though.

Daily quests would be nice, in fact I'd like the whole leveling experience to be better to be honest. Everyone just cheeses past it anyway as it is.

Relic weapon questline would be nice.

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I struggled to think how you'd do the collecting aspect of puppetmaster for a while, beastmaster would obviously tame a collection of beasts, but what would PUP do?

I think maybe an archeologist/researcher angle could work, master of all things constructs. You'd go and poke around ruins and stuff, plenty of them around, fight the various machina/golems/magical constructs within, and study the remains to figure out their secrets.
You could then build/salvage parts to construct puppets, which you would program with various compatible techniques.