1. What are the different ways to reduce the upkeep you pay every turn (not counting war-related expenses)? I know that you can build things like courthouses to reduce city maintenance, that having too many units incurs a cost, that having a religious shrine gives you +1 gold per city that has that religion, etc. However, I don't always get why sometimes the -4 gold I see next to my treasury suddenly goes to -2 without me doing consciously doing any thing about it. Can it have to do with trade routes or commerce (even though the % slider hasn't changed)? Are there any ways other than religious shrines (and if I understand them correctly, corporations, theoretically) to get a positive upkeep (+1 gold/turn or more) while having research at 100%? Also, I do know that the computer can change the slider itself if you're about to go in deficit...
The changing gold income can happen for many different reasons--trade route changes, maturing cottages, commerce multipliers, etc. Also remember that you pay more upkeep for a military unit
outside your borders than you do for one on home turf--so the amount of gold can also change if you bring a unit home.
DO NOT get attached to keeping the research slider at 100%. This is a common beginner mistake. Doing so will likely result in you stunting your civilization's growth at far too small a size; the AI loves to expand and will leave you in the dust with their bigger empires. By working cottages, claiming high-commerce resources, and running a few specialists, you can easily have 10 or more cities generating more research at 40-60% on the slider than 4 cities at 100%.
2. What is the effect of settling on a ressource? I may be dumb but I haven't managed to find an answer to this anywhere. Do you get access to its benefits without having to improve the tile (so that, for instance, I could build axemen by simply settling on a tile with copper)? Does it affect the food/commerce/hammer income from the city's tile? Or is it just making it invalid, making it an option only if it's truly an optimal placement?
Yes, if you settle on top of a resource you get the benefit of the resource without having to build the required improvement; it also makes the tile much harder for an enemy to pillage (since they have to capture the city to do so). However, you will not get the full benefit of improving the tile. Depending on the resource in question, you may get an extra slice of bread, hammer, or coin, but that's it. Therefore, you should consider the choice carefully. Settling on top of a food resource is almost always a bad move; food is vital in this game, and every slice of bread counts. Settling on top of a high commerce resource (dye, gems, gold, silver) is also counter-productive. Settling on top of a strategic resource (horses, copper, iron) is a judgment call--as I said above, pillage-proof, but you lose out on some hammers. Going this route is, therefore, situational--if you're in a tough neighbourhood with loony neighbours (Shaka, Monty, Genghis) and lots of barb-generating fog, it may be a good idea. Settling on top of the more mediocre-yield resources (silk, ivory, furs, wine, incense) is not a bad idea if it gives you better city placement.
3. Is there any way to get all units in a stack to attack at once? I feel like this is a silly question, but I don't have much experience with war (I'm a peaceful guy), so I wouldn't really know, and haven't found anything on this either.
Not exactly, but there is an option called stack attack which lets the AI decide which units to attack for you. As you get more familiar with warfare and the various promotions, however, you'll find you're much better off choosing each individual attacking unit.