#1 Just move on top of said resource in enemy territory.

#2 Civilization got every kind of building, Warfare horses and Fortress buildings, Magic has Conclave Spells and Essence buildings. What you need depends on what you have. There is no right answer for this, though maxing out research first is a good way to get everything  https://forums.elementalgame.com/447952/page/1/ this thread should have more advanced info if you need it.

#3 Bigger map. But eventually those militias will be too slow even though they are high level. Levelups is usually just about more HP, nothing else. I don't bother taking care of the early militias, and let them die quite often, and replace them with leather brethren asap. Then when I got horses/wargs, I start working on my end game army, so they get demoted to city defenders again. I play huge maps, so there usually is enough stuff to level up with, and if it's not, well the AI has plenty of fight to offer ^^

#4 I think there is supposed to be some visual cue, but I never use it. If you remove autoturn it will flash down right when you have spent all movement and all cities have something to do. With autoturn on (default), it should just go to next turn once you are done moving around. 

Sure it actually is a built mine? You can't destroy it completely if that is what you are after by just moving on top of it. I think you need to have it in your territory to do that. (if so, click on it and select raze, it needs to be unbuilt though, so you need to raze it before taking the outpost etc.)

Steal Spirit. (it's in your spellbook) Only works for magic paths and one or two special spells. So for instance if you get a really bad mage hero, and already have a good one, you can steal their fire magic for your sovereign. (Fire Disciple, so you still need to dedicate 4 levels to max it)

Regarding #1, if you move over a resource the enemy has made then you should hear the "Crash" sound and the build here icon should appear over the resource (assuming you have that enabled in options to see the hammer icons). 

At least on the difficulty I play on the AI immediately goes back to build on this tile though - even while having something else in the build queue.  I've ignored a unit sitting on an AI resource for a couple turns and noticed that the shrine or whatever had been rebuild while I sat on top of it, so I just moved off and back on and it was razed again.

I think if you walk over something then walk off, if you go back the immediate next turn it will crash it (raze it) again to be honest.   However, it's not completed that turn just the quequed to be built.

In regards to you first question #3, don't clear all the lairs... keep some that spawn weak troops (like bandits), they make great harvest for your newly trained upgraded troops to level them up. It becomes a particular art to create XP spawners (basically). I tend to leave darklings, or bandits just outside my borders so I can specifically train my troops up with their wandering mobs.

(I think the comp uses a similar strategy, if not, I'll just spread that rumor to make the computer look that much better).

As to your other questions, I believe your territory is sufficient to be able to open up your shop and I do believe that what you open with your shop in any city is available in any city (Although, I could be wrong here, never thought about the consequence of different areas in the territory. On a side note here, you can also shop in other peoples territory. So you may not have horses, but these other people might, food for thought)

As to mounts, they only increase the movement of the unit that has the mount. So, if you want a movement of 4 on the strategy map then you need all troops to have a movement of 4. But don't disregard mounts in general, on the tactical map that unit with 4 movement will still have 4 movement on that tactical map.

Usually, you don't want to. If the resource is attached to an outpost, you can capture the resource for yourself by simply moving a unit onto the outpost (which captures--not destroys--the outpost and all attached resources). If the resource is attached to a city and you plan on capturing the city, then the same applies. You'll get the resource for yourself when you capture the city.

As far as getting experience for later troops...

If you train all of your units in a developed fortress, they will start a few levels up to begin with. Also, there are usually harder monsters and quests available later for leveling (that were too difficult to take on early in the game).

No. mounts only improve the sovereign's move points.  So in tactical, your hero will move 4 (or whatever) instead of 2 but in strategic the stack is limited by the slowest lumberer in the group.   If you grab a horse with your sovereign and rally him forth with other mounted units then you can move at max speed.   Add one footsoldier and you are all slowed down again.  capeche?