We recently added account-level changes to achievements in World of Warcraft: Mists of Pandaria, and there's still a lot of work left before we're done. The plan here is to illustrate the intent behind our design, which should in turn help you figure out the answers to many of your questions.
Overall, we never want you to play Character A instead of Character B because of achievement concerns. If Character A had the Violet Proto-Drake, then you might not play Character B. If Character A was only one holiday away from the Violet Proto-Drake, then you may not play Character B. If Character A had completed most of the raid achievements from Dragon Soul, you may not want to bring Character B for one fight and miss out on the achievement. Having alts is cool and working on achievements is cool, but we don't want the two systems to work against each other.
This goal is paramount and drives everything else. If we allowed you to earn extra achievement points from completing an achievement on two different characters, then you might only want to play the character with the most points and you'd feel like you had to grind through all the achievements with every alt, thus defeating the purpose of having account-level achievements.
Most achievements are account-wide
This means you only earn the points once. If you have earned an achievement on one character, you can see it on all your characters. However, and this is important, you will still see the achievement toast (the pop-up notification) if a second or subsequent character completes the achievement. We think it's still important to recognize milestones like reaching level 80, maxing out a profession, or killing a raid boss for the first time. It's fun to have everyone congratulate you when you get the toast. Nonetheless, this will just be a new character of yours completing the achievement that you've already earned on your account. You won't double up on points.
Most criteria are not account-wide
If you start an achievement on one character, you can't then finish the achievement on another (there are exceptions, so please keep reading). We didn't think it made sense for one character to get level 60 and another to get level 20 and then see the "level 80" achievement toast. If you start to explore Thousand Needles on one character, you'll probably want to finish that achievement on the same character.
Some achievements are "meta achievements."
These are achievements that require you to get other specific achievements. An infamous example is "What a Long, Strange Trip It's Been." These achievements generally ARE account-wide. If one character earned Noble Gardener and another character earned The Flame Warden, then your account would get Long Strange Trip. (Otherwise it would be weird: it would look as if you had the sub-achievements, but you wouldn't have the meta-achievement). Even though you need Explore Thousand Needles on one character, all of your characters can contribute to Explore Kalimdor.
A very few achievements are account-only.
There are two categories of account-only achievements. One is achievements that are not possible to earn on one character. If we made an achievement to level every class to level 90, it would be account-only. (I'm not sure we will, but it's a good example.) The other category are achievements that are just brutal to complete on one character (and you'd never want to do for multiples), such as 2500 daily quests or 250K honorable kills. In these cases, the cumulative work of all your characters on those criteria will count.
We hope that rewards granted from achievements (pets, mounts, titles, and tabards) will be shared at the account level.
I'm not going to promise this yet, because a lot of magic has to happen for that to work, but it's our intent. We have discussed having a character level requirement for some rewards, so that your level-2 gnome couldn't walk around with the "Defender of a Shattered World" title just because your level-90 shaman earned it. If the gnome made it to level 80, though, you could proudly show off your title. You also won't be able to use a faction-specific pet, mount, or title on the wrong faction. The achievement wouldn't go away -- you just wouldn't be able to show it off on that character. There are probably additional exceptions and details we'll find as we dig deeper into the system.
This is the kind of system that will evolve over time, and we don't think this has to be the way achievements work forever. As beta players start experimenting with it and offering feedback, I am certain that we will end up making even more changes.
Greg "Ghostcrawler" Street is Lead Systems Designer for World of Warcraft. He loves you very, very much. Yes, you.