Normally my strings make it until I destroy my intonation and it's time for a set-up, so I don't normally have to worry about this...

I just blew out my high E string. Since the strings are in crappy shape, it's time for all of them to go. When I put the new strings on, do I have to put the new high E string on first and then go back to the low E string?
Nope. Any way is fine. But I would recommend puttin new strings on every month or every coupla months at the least.
Eh, I've heard of some people working from the sides towards the middle (stringing in an order like E,e, A,B, etc) and they say it makes a difference, but at one point, I would try that out every other month (I try to change my strings once a month) and I honestly could not tell a difference.
If you have a tremolo (whammy bar), change them one by one to keep tension on the springs.
^That really only applies to Floyd Roses or other floating bridges. For Strat type guitars it doesnt make any difference
You really should change your strings at least every other month, most people kill them after one month. You'll find it sounds much better, they are easier to play, they stay in tune better, and they break less often.

And yeah, some tremolos might get a little messed up if you change them all at once, and I've heard that you can mess up your set up (but I always change them all at once and have had no problem.)