1. As a healer you'll want to level destro and resto staves. You can level the others if you want. When you get to vet, you get daily enlightenment...400k ap earned at 4x normal rate. When I have a new vet character with an underleveled weapon, I do things like the world boss dailies, which net 100k+ ap. Turn in with a bar with that weapon and several skills equipped, and it'll level in a few days. Or grind the enlightenment w that weapon.

However, I prefer getting any skill line I know I'll want started.

2. In Morrowind Zos wanted to give players equal opportunity to start in Vvardenfell or the original game. The stories don't really overlap or spoil one another, so pick your poison.

You may do mage and fighters whenever you like. In the vanilla original game, there are several storylines: main story (via the prophet), a MAJOR storyline for each of the 3 factions, starting in the starter city and moving through all five zones, the mage and fighters guild stories. These all meet up chronologically when you open Coldharbour, and there are character and plots that intertwine a little.

3. Without any major spoilers, here's what you get roughly from these lines, in particular for your healer.

Mage: 2 skill points at the end, quest 6. But as a mag healer you'll probably want to level the line to 10, so might as well do the quests. The line levels by finding lore books, so the quests are entirely optional.

Fighters: not a lot for a magic character here. You level the line by closing dolmens and killing undead. You'll get there without doing the quests, they're totally optional.

Undaunted: this line won't open until your level 45. Its daily dungeons, not a storyline. The line levels via the daily quests and closing achievements on the dungeon section of your journal. Each undaunted dungeon contains 1-2 quests with skill points, so when you're high enough level you'll want to do these.

Dark bro and thieves: totally optional, but each main storyline quests has a skill point. This is a great source of points if you like playing the quests.

Wrothgar: some skill points in the main story, maybe 3-5.

Whether you want to do them as a vet or not is up to you. For me, before focusing on any of these I do the public dungeons. There are 20 in total, each with a sky shard and a group event skill point...so 26 2/3rd skill points all together...very helpful for a new character.