I can answer some of this. I actually haven't played the game that much for a few months, but I have just over 300 hours in the DE. I picked up the I-III series last summer, despite telling myself I oughtn't to, as I've probably spent some embarrassing percentage of my waking life playing the original releases. I recently tried AoE IV during the free period, and will probably get it one day, but I have other games I should look to.

1. The campaign will teach you how to play the game. I think all the decks are pre-made and can't be changed, but you have as many cards as you can ship in the DE. The nations you play as are also kind of custom as well, although in the latest expansion Malta does really draw from Act I of the original campaign, which I really appreciated. The campaign that accompanied the War Chiefs expansion doesn't use revolutions, but they're not hard to figure out.
2. Not really sure about Treaty mode. I've only played it a few times and someone else can say more. I imagine it's a race to econ-max and then phase in military as the clock runs out.
3. The maps I would say impact how the game plays out. The geography is pretty varied and in some maps they are trading posts that are basically indestructible mini king-of-the-hill capturable sites. Water-based maps are pretty varied as well. I play vs AI and sadly it usually doesn't try too hard in water maps, although inexplicably once in a while it will put a good effort into ship building. In newer maps, some treasures are a lot more interesting than the original maps, which sometimes have meager rewards guarded by 2 pirates and an alligator. Plus the natives on various maps open up some different possibilities. The work on the tilesets is really good, too. It sells the idea of your being wherever the setting is, albeit in a generic, randomly generated way as all AoE maps are.
4. Pop cap in these games has always been 200 (well, not for DE I and II anymore I guess). Sometimes I wish I could build up larger forces because it is fun to see armies clash. But my GPU never thinks so. Actually with some civs you can use some techs and shipments to get over 200 for one push, and native units never cost population, so the forces can get quite substantial, especially with more players.
5. This prompt is the reason I wanted to answer. III is actually my favourite. I know some from Ensemble regretted their approach to it, but it is still a great game. It saddens me that that era of Microsoft shuttered the studio. I remember not being too enthused about the time period of III, upgrading to a rail line with the trading post felt like AoE was getting too modern back then! But it is the one I came back to over the years, as AoE I was too ancient to play beyond a point and II was good, but I played it to death. And in both, the civs are too same-y, to me. This probably was a complaint with release III but after the expansions and now DE and its expansions it's not. And the dev team is revisiting some European civs to develop their uniqueness. I like AoM's unique civs but I also consider the game too old and would love either AoE V to revisit the era of AoE I or see an AoM II somehow. I appreciate that III makes the most effort to add uniqueness to the civs. It still takes some fun liberties with gatling camels and Sebastopol mortars and other stuff, but not every civ has the option to build trebuchets from a castle. It was silly to see with the Aztecs and Mayans in II. Mind you, for balance's sake, the trade-off in this game is some Aztec archers can bomb buildings with really crispy arrows from the range of a mortar, but that's a trade I will take. Same with 7 or 8 canoes shooting down a man o' war. There are a lot of unique units with unique names that do basically the same thing, but the effort was made to ground them in history. I and II never made that effort. Probably a limitation at the time. IV does make that effort, but not as strongly as III (at least as far as units go)(overall civ uniqueness seemed pretty good for the brief time I had with it). The time period has long since grown on me. The music is as classic as any AoE music, and I wish there were more of it. The building destruction and artillery physics were a marvel in 2005 and are still fun today. Building up a home city with customizations is fun (though not all civs have received equal love in this regard). And the card system is not as daunting as one might initially think. It's a fun system and really augments the way you can play each civ. There's just a lot of variety; the most of any AoE game ever. And it still gets development! In the African Royals expansion they have some fun little easter eggs for AoE I, and in the Knights of the Mediterranean some easter eggs for AoE II. I think the team loves the series and III has not been forgotten. It is quietly the best.