It sounds like you're looking for a restoration (healing) spec. The classes capable of healing are druid, monk, paladin, priest, and shaman. Racially there aren't a lot of significant differences, though specific races can only play certain classes. (Alliance races are listed below as well, though the original question was about Horde)
- Druid (Horde: tauren, troll; Alliance: night elf, worgen) - limited to leather armor; tree form (cooldown ability) increases healing throughput; good profession choices: skinning/leatherworking, herbalism/(inscription or alchemy), mining/jewelcrafting
- Monk (any except Horde: goblin; Alliance: worgen) - limited to leather armor; use brews and teas as the primary basis of skills; similar professions to druid (I've never played a monk healer, so can't really address the specifics...)
- Paladin (Horde: blood elf, tauren; Alliance: draenei, dwarf, human) - can train to use plate armor at level 40(?), limited to mail previously; some people think they are limited to single-target (tank) healing, but do have some raid utility too; professions: any of the combinations mentioned above, or mining/blacksmithing
- Priest (Horde: any except orc; Alliance: any) - limited to cloth armor; best professions: tailoring/enchanting or herbalism/alchemy
- Shaman (Horde: goblin, orc, pandaren, tauren, troll; Alliance: draenei, dwarf, pandaren) - can train to use mail armor at level 40, limited to leather previously; can also use shields; in addition to direct healing spells, also have totems that provide passive effects and healing; professions: similar to paladin
For all, you can learn first aid, cooking and fishing as secondary professions that will come in handy.
Note that resto is much easier to level as primary spec now that you get access to the LFD tool at level 15. And, they're in big enough demand that you shouldn't normally have queue times that are very long.
ETA: Cataclysm/Mists updates
This question gets a bit tricky to answer... Starting from scratch, if you buy just vanilla WoW, you can experience the original world as it has been destroyed by the Cataclysm expansion. You can level up to level 60, but cannot access professions, areas, races, or other such things from any of the other expansions.
In order to access Outland, jewelcrafting profession, and Draenei? / Blood Elf races and starting areas, and level 61-70, you will need vanilla WoW + the Burning Crusade expansion.
In order to access Northrend, Inscription profession, and Death Knight class, and level 71-80, you will need vanilla Wow, Burning Crusade expansion, and Wrath of the Lich King expansion. Oh, and a prior level 55+ character on your account to play as a Death Knight.
In order to access the misc Cataclysm level 80+ areas, Archeology secondary profession, Worgen / Goblin races, and level 81-85, you will need vanilla WoW, Burning Crusade expanion, Wrath of the Lich King expansion, and the Cataclysm expansion.
Those are the restrictions. You can play on whatever realm you want with whoever you wish. You just may not be able to reach all the areas they can.
You can trial every bit of it with the 10 day trial, though I don't quite know how they treat it when your time is up.
Best Answer
Yes. They do in fact carry over. If your subscription lapses your account reverts to a starter account, and you can play characters at or below level 20. Here's some more official information: https://us.battle.net/support/en/article/world-of-warcraft-starter-edition