In my opinion, Sanctuary isn't needed for any content, even Veteran raids. Its radius is quite small, and healing output without it is more than enough (the more that you run PvE content, you will start to see if people die it's largely due to a lack of understanding of mechanics/poor positioning). If you ever do find that there seems to not be enough healing to heal through certain "heal test" mechanics, I recommend that you see what you can change regarding your "rotation" in those specific moments, and also make sure people are standing where they should be in order to receive adequate healing.

Good 5 piece sets to have as a healer are: Olorime, Hircine, Worm, Hollowfang, Z'en, Martial Knowledge, Infallible Aether (if there is no Warden in the group post-Dragonhold), Mending, Jorvuld, and Torug's Pact. There might also be some niche situations where you could use sets like Powerful Assault, Morag Tong or Meritorious Service.

I'd say, though, for healing dungeons alone, you'd be set pairing Olorime with any of those, or maybe even with a set that allows you to contribute more damage (like False God, for example, or Mother's Sorrow, etc). In my opinion, there's no wrong answer for dungeons, especially pugs, as the "best" setup could very well depend on your group and the encounter.

Good monster sets to have are: Sentinel of Rkugamz, Symphony of Blades, Bogdan, and Earthgore. You might find other sets useful here and there, again, depending on the fight.

And for a staff, I highly recommend that you pick up the Master's Restoration Staff from Vet Dragonstar Arena as it will be the best option post-Dragonhold for offering your DDs sustain. If you find that it's not needed or maybe if you can't/don't feel like getting one, it doesn't really matter what you otherwise use.