Excuse me if this is in the wrong place, but I couldn't find a thread dedicated to simple questions.
This is pretty much it so you're good.
Can anyone explain to me how exiles work, and how to deal with them? I keep getting notifications of them ruining my improvements, but I never see them. Do I need a certain unit?
For most things, you can usually find info in the Player's Guide to v38.5 thread, or the information is linked to from there. But I'll try to more directly answer this for you.
It depends on whether you are playing with the Hide and Seek option or not. If you ARE, then the main thing they hide with is their Disguise ability, which is fairly weak but represents their ability to pretend to be friendly so you'll let them into your community and they effectively become invisible as they are a part of your population, or appear to be. Watcher units have the visibility capability to counter this but they have to be right up close to you to see them unless you've promoted your watchers to have stronger abilities to see through Disguise. Criminals like these may also have a little camouflage ability. Scouts are pretty good at seeing through that at first, and hunters get a little, and strike teams are very good at it but probably better to have out in the field, thus dogs are the best once you can get those, so usually a camo visibility specialist dog and a disguise visibility specialist Law Enforcement unit make for the cross reference you need in each city to spot most incoming criminals - until much later when they start using political invisibility methods and at that point you're on your own to evaluate how to counter that.
If they are ruining your improvements, they are likely hiding in your cities between turns - where a good investigation Law Enforcement specialized unit can help to successfully figure out who the perp has been causing all these troubles and either he or an anti-criminal combat specialist unit can then go in and make an arrest if they are still on the same tile after becoming 'wanted' thanks to a successful investigation check over the course of the turn initializing. They also become more visible when they become 'wanted'.
Without Hide and Seek it's a little more simple, only a matter of camouflage and you only have a 'yes these units can see that' or 'no they can't' factor to that so you only need to staff your city with any unit that can see through camo.
Also, what's the difference between stone spear/axe/mace men? I tried using the ctrl button to bring up information about them, but it does nothing. I've ctrl + left mouse click, holding ctrl, and just pressing ctrl.
When you press control, do it before you hover over the unit or unit icon. If you're trying to hover THEN you hit ctrl, it doesn't change the display - the tooltip display has to initialize with the ctrl already held down.
Spears are good against mounted, also against animals in the early forms.
Axes are good against melee.
Maces are a little good against melee and fairly good at capturing cities. I still plan to give them some city defense eventually as well.
Swords are very good at capturing cities, but not as good as a mace's modifiers IF the defender is a melee unit.
Throwing units are good at guarding AND capturing cities and are particularly good against archery units. They get better at attacking cities and worse at defending them the more advanced they become through upgrades. They get decent withdrawal capabilities as well.
Archery is very good at defending cities. They are also the only really early units that can distance attack.
Mounted is good against throwing but bad at attacking cities somewhat. Their very strong withdraw can make it worthwhile still.