As far as addons, I currently use healbot and have been using healbot for the better part of a decade. However, recently it's been bugging me that it doesn't actually show me all the available information within raid frames. And since I had been effectively using it as a raid frame replacement for my healers, I now find myself having to use raid frames to follow many items that don't appear on healbot's UI. The easiest example off the top of my head is Xavius. I don't actually need to see the number of stacks the Boss tank is receiving in phase 1 considering we have the tanks call out when they're ready for the dispel. But I feel that my ability to properly assess the current situation is hindered without knowing exactly how many stacks the tank has and when. I also wouldn't mind seeing everyone that's been put into the dream throughout the fight. It's not necessary for healing per say, but it's a nice bit of info to have on hand when you're one of the voice trying to help coach the raid through the fight. So instead I think I'll be looking into a click-cast addon that allows me to cast directly on to my raid frames instead of replacing them with something else as I currently am.
In general I bind my offensive abilities, mana pots, trinket "Use" abilities, personal defensive's and CC to hotkeys. I don't know how others bind, and my actual method for keybinding might be a hindrance to some, but I adopted it from my tank keybind layout. It's pretty much a cascading 1, 2, 3, 4 ; Shift-1, s2, s3, s4; s-Q, s-W, s-E, s-R; s-A, s-S, s-D, s-F; s-Z, s-X, s-C, s-V. If you look at these key groupings on the keyboard, you'll see what I mean when I use the term "cascading" to describe it. I put all my offensive damaging abilities on 1-4, and then find a comfortable spread for the rest.
I do my healing exclusively through click-casting with Healbot. Shield, Purify, Please, Radiance, Shadow Mend, and even Pain Supression I have all set to trigger on click.
This organizes my healing and other spells and abilities in a way that makes sense to me. But something like this is particularly a personal choice.
As far as talents, I have two particularly different approaches to Raid healing vs 5-man healing. I'm not claiming to be some authority on Disc Priests, but I have found each approach to be suited for their intended purposes and catered to the stresses of both. I certainly have some refining to do but another perspective can't hurt.
As a general strat for 5 -mans I focus on Atonement healing via DPS throughput. As a result of this approach, I choose the following talents:
Schism - That's a 30% bonus to damage on the selected target. That equates directly to a 30% increase to atonement healing from DPSing that target while the debuff is active.
Angelic Feather - For those times when you and/or a few people are lagging behind the tank, it's a quick easy way to get 3 guys caught up quickly, or get yourself caught up if you've fallen well behind for whatever reason. The cooldown is long, but you can chain 3 charges in a row, which I prefer.
Dominent Mind - I truly wish there was more use of CC this XPac. I genuinely expected there to be, but because Mythic+ is all "FASTER FASTER FASTER", it's use is pretty limited. Despite the trend, I still spec for the extra CC ability in case it's ever needed.
Mindbender - I don't know whether or not Shield Discipline would offer a better return on Mana in a 5-man with this strategy than Mindbender (pretty sure it does in a Raid, however). But regardless of that, Mindbender also offers a small increase to DPS when it is active, and turning that 3 minutes CD into a 1 minute CD with a mana tick is pretty dope on top.
Power Infusion - Some would argue for Twist of Fate here, even with this setup. But I prefer to be able to control precisely when it is that I'm going into "oh shit ohSHIT OHSHIT!!" mode. The 2 Minute CD is the worst thing about this. The amount of burst healing you can get out of Power Infusion is fucking MASSIVE, but the catch is that it's not going to pop if your group is near-wiping on every other pull. Twist of Fate is a good way to go if you know you're tank or DPS are going to be near death constantly.
Divine Star - I don't use this as often as I should. The little HPS and DPS explosion it causes is moderately noticeable. Not fantastic, but pretty decent enough. Halo is a definite no-no for 5-mans. It is an agro nightmare, potentially worse than a hunter's barrage. Clarity of Will, I just already have too many spells now with a cast timer, adding another I believe to be a hindrance. Divine Star is instant cast.
Purge the Wicked - Instant cast. Better DPS throughput. Works great with Power Infusion. A lot of fun to spread around to a bunch of little shits for some lazy healing, epic DOT/HOTage. A must for this approach. Grace is horrible for this strategy because it focuses on direct heals, which this strat generally avoids. Shadow covenant... heal absorb? ick. No way Jose'.
Te general idea behind using this strat is high uptime on atonements and heavy offensive focus. Purge the Wicked should be used liberally and spread most often using Penance, which should effectively be used on CD after your atonements and Purge the Wicked are out. Even if you need to get health up, Purge the Wicked does an immediate tick of damage/healing so poping it just before Penance is usually ok... unless the health is that low, in which case get that Penance out quickly. Use Shadow Mend, but use it sparingly. Only when health pools are starting to tick a bit low and you don't have a bunch of atonements out already. If you need to apply atonements quickly but health pools are currently ok (like in a situation where you expect a bunch of damage to start soon) you can use Plea to pop out 1 or 2 atonements, or even use Power Word: Radiance to get out a quick 3. Power Word: Shield should be used almost exclusively on the tank on CD, unless it is the only thing you can do to save someone else from dying. If you only need to get one atonement out while 4 are already active, I would typically use Shadow Mend instead of Plea because of the increased Mana cost to Plea. But if you need it out that quickly, Plea isn't bad.
So a typical pre-pull set-up is to pop a shield on the tank, maybe toss out another atonement or two, and start DPSing. Throw out atonements as they are needed and keep your DPS up. Use Shadowmend to spot heal when necessary, but try to rely mostly on your DPS abilities.
Now when it comes to an "OHSHIT" situation, I have a pretty typical go-to cast order. I almost always have at least 2 atonements out during the fight, for context. Power Infusion / Mindbender -> Shadow Mend (if needed) / Power Word: Radiance (if the atonements need to go out NOW) -> Schism (if you can) / Penance (if you can't Schism first) -> Light's Wrath (boosted because of the now 5 atonements out) -> then alternating Smite and Penance on CD on whatever mob is causing the most single target damage (ideally). Still use Shadow Mend to spot heal if necessary and keep that shield on that tank unless otherwise needed.
Also, make sure you have access to Mana Potions. I prefer the generic Ancient Mana Potion for 5-man content mostly because I don't usually have that much trouble with mana in Mythics... typically. But I do have my mana pots as well as a crafted 850 mana Neck that I keep on, for those situations where it's necessary.
I didn't expect to write this much, so I'm going to stop here. If you really want it, I'll go over my approach to Raid healing, but only if you have genuine interest in it. Again, I don't claim that my approach is any better than anyone elses, it's just what I've found to currently works well for me.