Got the rules, got the faq's for the rules, got the general's handbook and the errata for that too, but I must be a bit Tom and Dick cause I still have the odd question or two...
Welcome, and don't worry, we all start off with questions, and need a few games, and few reads of the rules before we really get ir right. And even then we forget things now and again.
Starter for 10: When two units fight, is it only those models with melee weapons in range that get to attack (no missile weapons in this example)?
Yes. Missile Weapons on the profile are only used in the Shooting phase, and Melee Weapons are only used in the Combat Phase.
Also check the description section of the warscroll, as although you're allowed to use all the Melee Weapons on the profile, that does depend on what they are armed with. For example, a Demigryph Knights unit can either be equipped with Lance and Sword, or Cavalry Halberd. And that's all the models in the unit in most cases. There are a handful of units that can pick and choose per model. So if you were equipped with Cavalry Halberds, in the combat phase you use the Cavalry Halberd and Demigryph's Razor Sharp Talon, but not the Lance and Sword.
Many of the monsters and behemoths can have 3, 4 or even 5 attacks sometimes. Again, the description could limit some choices.
If so, and the player losing models gets to choose which models to remove, can they pick any from the unit, or only any from the models in contact?
Any models from the unit. If a model has a wound taken already, then that model must suffer wounds until it is slain before any others are chosen.
Next Q: Friendly side has two units A and B in close proximity to each other when an enemy unit C attacks (their turn). C attacks B and slays half the unit. B then attacks but doesn't wipe out C. A is in range to pile in and can (according to the FAQs), so does A attack first? Does C even get to attack A?
Every unit on the table can only be activated once per combat phase. The order they are activated is up to each player, starting with the player whose turn it is.
In your example your opponent activates his unit C in his turn first. Unit C has most of it's models in close enough range, they have to pile-in toward the nearest enemy in unit A, however, some are closer to unit B, so can pile-in toward unit B - as long as all the models in Unit C are still within an inch of each other.
Every model in Unit C can then attack a unit within range of it's melee weapons. Where the model is range of 2 or more units, the model must declare which of it's attacks are attacking which unit before any dice are rolled.
So unit C in it's turn could attack both units A and B.
Having done so, and these are the only units on the table that can be activated, you could activate unit A or Unit B. If there were other combats going on, your opponent could choose to activate unit D for example. But if he only has unit C, then he has finished, however, you can still activate all your remaining units, so B could then activate.
You might like to consider if you attack with A, and your opponent removes models at the end of their unit facing B, then B could enough up out of range of pile-in and then couldn't activate. As you asked about removing models, this is one way that can be used tactically, though it's not always they best option, as those models might be the best to face unit B anyway.
More later...
Cool, ask away