I never know whether to go for say Robotics for Mech Infantry or the Internet.
Mech infantry you CAN get later. Internet, once you miss that race, you CAN'T get. Internet first. Then mechies. Although leverage your Radio while building Internet, for Bombers, which are higher priority than mechs, and for Cristo if you're not Spiritual (civic dancing late-game can give a huge boost for when you're not always at war nor always at peace!) Even if they can intercept, you never regret flooding a battlefield with bombers early and often, ever.
I don't really know how to use spies (what's with the cuirassers + spy strategy - how does that work?)
If you keep them on roads they move at the same pace. When they get to the city, the spies become "human siege engines" and whack the city defenses sorta like cats/trebs/cannons. The remaining attack is done by the cuirassiers. Highly mobile, highly deadly. IF you're generating enough EPs to keep up the pace that such a deadly tactic will bring, and don't forget to race CGs into the cities you take (along the roads that open up after the conquest) because you don't want to defend with just wounded cuirassiers.
he started stealing everything! Everytime I'd get something I could swap with him - he'd rip it off the next turn! How do I stop this mid-game? Put spies in every city?
That plus counterespionage missions by your spies.
I know there are some late game resources like Oil and Aluminum that are critical - but how critical?
Critical enough that if you don't have them, job 1 is now to build Aluminum co and Standard Ethanol to get them. And until you DO get them, by conquest, you're stuck in Free Market, no State Property for yoooo!
What about corporations? Worth saving a GP for?
IF you have the resources, to make the bonuses worth the effort. If you have the metals and coal, Mining, inc. is just a *sick* production boost. If you have the seafood and/or rice, Sid's Sushi can eliminate the need to choose between CE and SE and allow *BOTH* (that much food!)
I used to avoid corps if I was planning on a Dom/conquest victory overseas, but a game I'm playing right now I've stayed Free Market, and even spreading corps overseas isn't a bad hit when the hq of the corp is in the wall street city, and once you get a courthouse put in (1 net gpt cost after Wall Street modifiers, for +17 hammers? YES please!) Closer to home they can even turn a profit, like a "late game shrine" effect.
How and when do I go for the kill? I know Combustion is big for transports and destroyers - just like astronomy is for galleons. These are big mid-late game military techs (assuming probable naval invasion if planned).
Depends on what navy the AI's got and if you can get the drop on them. If they're cruising the seas with battleships and for whatever reason you want to land cavalry for an invasion with a wood ship navy, obviously that'll be the first last and only time you'll make the trip. If you've also got Flight that might not be so bad (airlift becomes your new transport method after that first couple of cities), but... plan carefully.
It's not uncommon that some backward aggressive AI invades me from overseas landing knights to fight my cavalry, with wood ships that are crunchy and good with ketchup for my ironclads or destroyers. At the most they get 1 or 2 tile pillages in before they're converted to compost. My advise may be unpopular but, don't follow their example.
And what's the deal with muskets? I've never really used them offensively - they didn't seem to be any better than a decently promoted maceman. Am I missing something?
They're just versatile is all. They can defend a stack reasonably well except versus knights (supplement with PIKES of course), and on flat land they squeak out ahead of longbows for city defense (with comparable CG+drill promos). Supplement city D with pikes too, naturally.
I also like the "be prepared to play from behind" - for a lot of the game. Be prepared to defend. Send units to borders cities. You don't need 8 longbowmen in your capital (unless required for happiness under HR).
If required for happiness under HR, stack the 8 obsolete archers that were replaced by LBs in the border cities once you got Feudal with it. Along the border let the full stacks in city D be up to date units, and let the obsolete ones focus on riot control at the center of the empire.
Maybe someone else has got a better idea but I like to have 1 unit in a city (unless more reqd for happiness), unless it's a border city where I will have 2-4 decent defenders, and then everyone else in one location ready to go, preferably in the military production city assuming this is centrally located for defense.
Border cities need to have enough defenders that the AI units you see in range will not be able to rush ride immediately to them and overwhelm them. In other words the level of defenses is driven by how many units your spies or scouts are reporting back. If it's a peacemongering AI who forgot what a sword looked like in ancient times, you can probably get by with 2 archer types and a spear type, in the cities themselves. If it's an aggro like Monty, you know what, even if you see NOTHING with all your spying and scouting, beef up those cities as much as you can afford to anyway. Because it's Monty!
A little bit back from the border cities should be a mobile defender stack ready to run to a regional border city and counterattack an invading stack. This force will be mostly horse-types (HA, Knight, Cuirassier, Cavalry), but include melee/combat types (maces, muskets, crossbows, whatever) and siege promoted on the drill line, for those cases where an invading stack isn't siege heavy. (Again, MONTY! He'll send 30 of just knights, ain't no thing but a chicken wing, to him!) In fact if it's Monty and he's got horses, or Genghis, increase the level of spears/pikes in the defensive stacks. Mobility is important here because this coverage defender stack has a "zone defense" duty. Each border city should have a defender stack within 2 turns march by road. Wider frontiers will require more defenders, which is why it's strategically important to plan for "choke points" in city placement when possible.
Also - when you talk about Bronze Working you mention chopping, but BW gives you 3 huge things early on: ability to chop, ability to whip (switch to Slavery as you say) and your first critical military resource location. That's why it's a big deal. I know you mention all of these in various places, but I thought it was worth pointing out, because it's so critical to early game strategy. When you hit BW have workers poised in place to start chopping, switch to slav and start whipping, have workers ready to go and hook it up asap to all cities - have a settler ready to go found a copper city if it doesn't pop in your borders. If you don't have it... better have horses and/or start looking for iron asap - maybe both!
Although if you're Monty or S Bull you have other priorities--the horses become more important right off the bat, and then iron for later units like xbows, cannons. Or if you're Cyrus and you plan to rush, definitely horses.
You DO WANT BW anyway for the whip if nothing else, but the city placement priority is what shifts around with those leaders.
I don't do a lot of this - which isn't to say it's not a good idea, however what I do do now is go to 0% science periodically for the purposes of military upgrade. Might be worth sacrificing a few turns of research to get all your archers to longbows, axes to maces or whatever. Don't spend the money carrying outdated units that you'll never use. Obviously the usual caveats apply here, but I find myself doing a lot more: stopping research to build up gold, upgrading units (higher promoted, in border cities first) and sometimes deleting old units. Keep that army a lean, mean fighting machine - not an overbloated, outdated, expensive bunch of underpromoted troops scattered everywhere but where you need 'em.
To me the cost of upgrade is only worth it if the unit has built up enough promotions to make it worth the significant gold cost (relative to the lower hammers cost of just building a new damn updated unit). If I had archers sitting in a back city, build in pre-vassalage times at CGI, there's no way in hell I'll spend the gold to make a CGI longbow out of it. If I'm running HR that archer is now a policeman. If I'm not running HR that archer is now "retired" (after replacement with a newly-build longbow). Of course, if it's an archer who fought city defense in a disputed city and is now CGI, DIII, yeah, that's a keeper. Upgrade. All in all my rule of thumb is: if I can build a unit fresh at x promotion level, DO NOT spend the gold to upgrade those units. Build-replace instead. If a unit has a higher promo level than what I can build fresh, *consider* the upgrade (assuming it's not outdated promos like Shock in the gunpowder era). And of course CR-ling maces, I still love to upgrade those to grens/rifles/infantry after CRIII, no matter what anyone says.
), are you sure it's not worth it to wait for the dam, even if you beeline plastics, and have an engineer and US ready to drop it immediately? I mean, of course you get many extra hammers by building plants earlier, but think of the hammers you save if 10+ cities don't have to spend 5-15 turns on a coalplant in the first place. 1 coal plant = 1 marine.
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output?? 
Even if you lose some population, it doesn't hurt much. Granaries keep losses slow until you get the healthiness back up.
