For the most part, the BC and Wrath content is still available. It’s just out of order right now because of the 1-60 revamp in Cataclysm. Once you get to 58, you’ll get a pointer quest to choose between BC and Wrath. (There’s nothing stopping you from doing both besides the fact that you will outlevel them long before you are done.)
There are way way more quests in the game than needed to level. Doing all of them will take a very long time. For the most part, zones will have their own stories to them. They may tie into the bigger picture of an expansion, but are largely self contained. (This is a bit less true for the BC and Wrath zones which are a bit more free-form.) The best way to approach the story is to think of it as a collection of stories that aren’t always in chronological order.
There are 4 main options for people who want to do all the quests. (And you can mix and match between them.)
(1) Do different zones on different characters. This is probably the most popular choice as it’s the most efficient and makes leveling new characters a bit more interesting as you get to see different stuff.
(2) Move on to new expansions when you’re level is ready and go back and blow through the stuff you missed at max level. This has the advantage of letting you do the max level content when it’s current, while filling in the old quests when you have time. (There’s a toy vendor that wanders around Orgrimmar and Stormwind that sells a “Soft Foam Sword” toy which can be used by high level players who will one shot stuff for quests where you don’t want to actually kill the target.)
(3) You can keep questing even after you outlevel an area. You’ll start getting an XP penalty when you hit a zones cap +1. (So, in the 1-60 content, you’ll start getting the penalty at 61.) It’s starts off mild, but rapidly takes the XP down to trivial amounts as you level. This is not efficient, and it will take you a long time. (But if you’re having fun, go for it!) You won’t really overpower stuff until you get a fair bit above it. (For 1-60 content, the late 60s.) After that, most of the quest objectives will start to become trivial, and you’ll also be able to start soloing the dungeons.
(4) Basically, the same as 3, but you lock your XP at the expansion cap levels. There’s an NPC that will let you turn off XP gains for a bit of gold. (You can talk to the same NPC to turn it back on when you’re ready.) So, when you hit 60, you’d lock your XP and then finish up the 1-60 stuff. This has the advantage that stuff won’t get trivial and you can still queue up for dungeons. (Once you outlevel a dungeon, you can’t queue up and can only enter it manually.)