My own opinions:
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Are there any significant changes I should make to my playstyle, spec, or reforging?
From my POV, I changed from 25 to 10 man. In 25man, there's less single target sniping you need/can to do because there are other healers that can do it better than us (RShaman, Hpally). You would most of the time focus on AOE healing. From what I'm seeing, you are not using your Prayer of Mending as much as it should be. Hopefully you are using it on cooldown because it should account for quite a bit of your healing %. Make sure you at least use it on high AOE phase like Hagara Ice phase stacked, Zonozz Black phase, Madness cataclysm etc any AOE aura type of damage.
Body and Soul is very situational in most cases. With your guild's progression (8/8 on farm?), ice phases should be quick and not much bloods will need to be kited on Spine with the 20% nerf. I am actually the same as you, I switch between Holy/Disc/Shadow so there's not much reforges I can do without spending time & 1k gold+ every week to respec and reforge. Reforges shouldn't affect your throughput much anyway. If you have too much mana I suggest you spec into Darkness instead for the extra haste.
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How bad is it to have 2 Disc Priests and no Holy Priests in our comp?
2 Disc is not terrible for 25man but should be the maximum amount there should be due to Rapture mechanic. However with your raid comp only consisting 1 Resto druid, you might want to prefer Holy for an extra Divine Hymn/Tranq on most fights.
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What are some useful Lightwell addons?
I use an addon called Raeli's Spell Announcer (RSA). It can announce that you have dropped the lightwell, track the number of charges left as well as who clicked it. Technically your raid doesn't have to click your Lightwell whenever they take damage. How I get my raid to click my lightwell is that I only remind them to cilck it on high AOE phases like Spine 9 stack explosion, rolls, Madness Blistering tentacles etc high damage phases. Make sure you remind them and let them know the location of your lighwell. Because most of the time they'll tell you I CAN'T FIND IT FFFF.
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Chakra Serenity vs Chakra Sanctuary:
Sanctuary all the way unless you do the Serenity renew on Spine. I don't see any other reason to give up Sanctuary on 25.
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Is it common to have healing assignments where healers are assigned to a specific group to heal for AoE healing?
That will have to depend on your healing partners. Are some of the healers new that you've barely healed with? Or old guildies that already know whats gonna happen. Some groups you can assign for the first time and they'll remember what to do. If you're in the guilds that swap healers all the time it's probably a good thing to assign healing cd's.
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Random gear question: Should I continue to use H Bracers of the Banished for Holy?
Depends if you need the spirit really. You can always use the Banish on farm fights and use the spirit ones on spine/madness where you actually need the spirit for longevity.
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Specific question: Is Archangel worth having as Holy on Madness?
Hmm I think if there's much down time on Madness you should drop a healer
There's always something to heal, tentacle slams, blisterings, cataclysm, 4th platformomg
Also I do think that you could drop a healer for some of those farm fights as the overheal % of those fights are really high (45%-70%). It's hard to see performance when all the healers are doing tremendous overhealing
I must be super bored to type all this.