1.) You have a defensive tool, it is "Horrid Roar" (10y radius, -50% dmg vs you on enemy in it). As a Dragoon you are more than well equipped as a hit-and-run bruiser DPS. If you don't hit enough or all enemies, that is a different issue. If you get stunned / CCed on doing High Jump -> Horrid Roar, then you Purify -> Roar -> Heal/Chaotic Spring for sustain then continue your burst.
If you run out of Roar uptime, you fill the gap with Guard where necessary and Elusive Jump out where it is practical.
2+3.) If you say you are the initiator as a Dragoon, you are more than well-equipped. Doozer's comment is partially correct, as the need for Recup spamming depends on the map layout/location and the timing of your engagement. I use corners/walls a lot in my engagements to minimise the LOS from enemies on me and play around creating and holding space rather than diving hard into the open.
It is also not fair to say that ending up not healing a whole lot equating to activity. Many factors come into play on when you get focused hard (and thus need to use Recuperate proportionally more).
The line between being in a teamfight and overextending a bit too much is very narrow. I am not saying you are diving on your own, I am saying you may overextend. It is arguably more important to manage your own ressources as well keeping track of what the enemy has (MP, recent Guard uses, recent Purify uses, recent crowd control/defensive uses) than ensuring a kill as soon as possible.
That said, you are spot on saying that combat engagements are essentially fast-paced and depending on jobs played a bit hit-and-run. Whether you like it is a different story, but as it stands this is the current design philosophy.
I have been an advocate for the return of CC resistance (10s immunity after CC based on type) and/or at very least the return of Fetter Ward (CC immunity as a buff) as a global action to incentivize pushing without the fear of being blasted by CC the moment you press Jump.