Dual-boxing is a great way to try out the idea for a month for only the $9.99 that wow currently costs. There is plenty of scope for exploring the hobby from 1-60, whether with questing pairs, dpsing in dungeons (please only do this once you are competent in the open world!), or even running a tank/heal combination in dungeons. I personally have only dual boxed this expansion, except right now I am doing a 5 box with 4 "free" old veteran accounts mostly as a proof of concept.

As of today you don't have to pay the $75 a month. But yes, plenty of people do. It depends on how you look at your hobbies. Many people think that $75 is a reasonable amount to pay monthly for a hobby. Most console gamers, for example, spend that to buy one or two games a month. Most drinkers could spend that in one night. All about priorities. You certainly do not have to buy 5x copies of Draenor (I personally haven't), but there are people that run 10 boxes and find that worth it.

I think that shaman are commonly the most popular class to start with. Ranged + with heals (that dont break form like shadowform for priests). Moonkins have an annoying eclipse mechanic that even experienced boxers find a PITA to deal with.

As for your first healer/tank pair, I'd go with prot paladin, disc priest. Welcome to the community, don't forget to check out the IRC chat rooms if you get stuck and can't find an answer to a problem after an honest attempt to do so.