The lodestone has a page showing what the services unlock: https://na.finalfantasyxiv.com/tales...ure/#mainquest
Click over to the Tales of Adventure : A Realm Reborn and Heavensward tab to see what it includes.
You can click the green Available Trials and Duties header to see a complete list for those.

For the things you mentioned specifically, PVP is not unlocked, mounts are unlocked and you will automatically be attuned for flying in HW zones.