...and I can't remember if this has been done before, but is it possible to limit the number of loads/unloads of a unit? Otherwise, if you place your ships wisely, you can cross half the map in 1 turn!
It hasn't been done but it is possible. I already limited units to one airdrop per turn so that, if you have airdrops configured not to end units' turns, they can't keep airdropping across half the map. I could do something similar to limit units to loading once per turn.
1) I noticed that when using "disallow trespassing" function, helicopter can still drop any visible units into enemy territory, and they can even continue to walk there until the AI require to move them away. The same situation is with paratroopers, they can jump into the territory of another player (with whom there is neither a war nor a right of passage treaty) across the border. Is this a bug or a feature?
I never thought of helicopters or airdrops when working on the trespassing restriction but it doesn't make sense for them to bypass it, so I'd consider that a bug. The fact that units can walk around within foreign territory as long as they manage to cross the border is deliberate, though. The alternative would be preventing them from moving but then they wouldn't be able to leave. Technically I could bounce them out of foreign territory at the beginning of the turn like Civ 4 does, but that's a lot more effort. Something to think about for the future.
2) Also, auto-recon function is very much needed for air units. If it was possible to implement the stack bombing function, then I thought that it would also be possible to implement auto-recon.
Auto-recon would be possible, yes, though not easy. One thing I'm not sure about is when during the turn the auto-recon should activate. You'd probably want it early in the turn so that you could move other units based on the info revealed by the recon but if it goes off too early then the units spend their moves and you lose the option to give them a different order until the next turn. Maybe the simplest thing would be to make auto-recon activate at the same time as auto-bombard, since that way they're consistent, but I'm not sure if that's useful as I never use auto-bombard myself.
3) Re-base distance now is defined as x6 the plane's operational range. This is a little bit unrealistic because in reality it is x2 or x3 max. Of course everyone may have their own opinion on this matter, but would it be possible to make this value changeable?
This is possible, yes. It wouldn't be as easy as overwriting a number because the factor of six is baked into the logic and then the logic is duplicated, once for human players and once for the AI. Still, it wouldn't be especially hard.
4) Is there a way to make precision bombing truly precise, so that if you know what buildings are in the city, a window opens with a choice of a specific target (just like with a stealth attack against units)?
With enough effort anything is possible. This might require a lot. Popping up a window with buildings to target would be easy but then modifying the precision strike logic to target the selected building might not be. I don't know what that would entail.
5) Flag "Replaces all Impr. with this Flag Checked" - is it possible to somehow reanimate it, so that it would be possible to separate the types of buildings that must replace each other?
I looked into this a while back and am left with the impression that it's unfortunately pretty awkward to change. One of the problems I remember is that the AI's logic for choosing production assumes that buildings with the replaces-others flag are powerplants. It considers replacing those buildings based only on production bonus, pollution, and maintenance cost. I didn't see any easy way to modify that, and if the AI doesn't handle building replacements properly it can get stuck in loops of replacing buildings back and forth.
6) Is it possible to make water tiles give access to any start/lux resource located on them like roads? In this way it would be possible to make very elegantly, for example, an empty and useless strat.res. offshore oil, to which the city would have access via the sea automatically, and which would be required for the offshore platform construction, which in turn could generate regular oil (thanks to your mod) and be built only in a city which has the offshore oil in it's radius (using the corresponding flag in the editor).
Maybe. I could try assigning them to the same road network as the nearest city each time the game recomputes the road networks. But if all you want to do is set up a building to create oil from an offshore source, you can do that already. You'd make the sea oil resource a bonus resource then have a building required by the oil platform, like a harbor, generate sea oil locally. Then make the oil platform would require sea oil in range and generate regular oil. The only catch is that you need to use Quintillus' editor to assign the bonus sea oil resource as a requirement of the oil platform because the Firaxis editor doesn't let you assign bonus resources as building reqs. Another advantage to making sea oil a bonus resource is that it doesn't show up on the interface or become available for trade.
7) a) A loaded air transport should not be able to rebase, otherwise it breaks the significance of its operational range.
b) Loading into any transport from city, the unit does not spend a turn, so theoretically a unit can fly and sail around the world several times in one turn.
This is another reason to restrict units to loading once per turn.
8) I haven't noticed this before, but it turns out (thanks @tjs282) that the capital never shows as a size1 city, making it impossible to recognize if another civilization's capital has walls until you can investigate city. This is weird to say the least, is there a way to fix this?
Yes, it would be simple to modify the city drawing logic so that there's nothing special about capital cities.
P.S. Your work is really incredible, so many important changes!
Glad to hear it.

There's more on the way!
- Is it possible to limit bombardments to sea or land tiles only? That would make torpedo or napalm bombers possible (the attack animations already exist).
Yes, I believe this wouldn't be too difficult. There should only be two functions I need to modify, one that judges which tiles the human player may bombard and another for the AI. (It's annoying how often there are two separate functions in these cases when only one is needed.) The hardest part would be reading the rule in from the config file.
- just bringing up an older question (maybe for later)
I'll try to get around to this at some point. I'm a bit reluctant because, for example, "[all] Class CA" is a valid unit name on its own, so what would you do if you wanted to specify that name without invoking the wildcard? A more typical wildcard would be "*" however that's allowed in unit names too, along with all the symbols I can easily type. Not sure what the best solution is. I'm thinking I could either create a special kind of text, for example *"..." would match wildcards inside the quotation marks but plain "..." wouldn't, or I could have the wildcard be specified by another config option. That would push the problem onto other modders and make them choose a wildcard that doesn't conflict with any of their unit names. Anyway, not sure when I'd get around to this.
Any luck getting us an option to allow us to use era specific irrigation and mine textures?
It would be easy to swap out the sprite sheets when drawing irrigation and mines, unless there's some kind of special way they have to be loaded, which I doubt. Again, not sure when I'd get around to this but I'll keep it in mind.