Do I need to have a separate copy of the expansions for each account?

I have the expansions downloaded from a previous account, but want to use a different account to play the expansions on.

Is this possible to do this without buying them again? If I have to buy it again, and I bought the digital version, could I switch computers sometime and play my account on that computer?

You need to purchase a key for each account you want to use an expansion on: if you have two accounts, you need four keys: two WotLK keys and two Cataclysm keys. Simply downloading and installing the expansions won't enable those expansions on your account.

However, you don't need to re-download the expansions just to use a different, expansion-enabled account: as long as you've downloaded and installed them once on your computer, you can use any account that has the expansions activated on it.

As long as you have the expansions installed (and a valid account) you can use them to play.

Think of the accounts and the game discs as two different things.

The game discs buy you "the world", while the account (and monthly subscription) buy you access to that world.

One thing I'm not so sure about is if you take your account with a lvl 85 char and try to play on a computer with only vanilla WoW installed. I guess you wouldn't be able to log ind, but it's not something I've tried.

The content is linked to your WoW account (NOT your battle.net account, if you have multiple WoW accounts on a single battle.net account each needs to have its own purchased copy of the content as well as time).
If you have Cata installed on a computer and log in with an account that doesn't have Cata access, you will have access only to the features available to that account's content level. If you log in with an account that has Cata access but no Cata installed, it will download the remaining content on the fly.

It depends on where in the world you are playing.

In Korea you only need the game installed and you can play all content.

But for the rest of the world you account need a cd-key registered for each expansion to play the content that apply to that expansion. But you only need to have the game installed one time, if the account you are playing do not have all expansions activated then you just can not accesses the content from that expansion.