The answer to most of your question boils down to a problem with one simple statement:
They should work together, and they do
If they did, then we wouldn't be having this conversation. Just because two mods (or a mod and a scenario, or a mod and a map script, or a out-of-date mod and the updated version of the core game) don't immediately crash your game and cause your computer to burst into flames doesn't mean they actually do work together. Incompatibilities can be far more subtle than that, and in fact usually are. Yes, you might be able to go a little ways in-game before you see anything noticeably wrong, but that's not the same as saying they worked together. It'd be like saying that if my car's engine explodes when I go above 60mph, then it was working fine for several weeks of local driving up until it went boom; no, it was broken all along, you just didn't see the effects of its brokenness until you tried going on the highway.
Any mods that try to edit the same UI window will not be compatible, period. Likewise, any two mods that add new unit graphics or alter existing ones are not compatible. Two mods or map scripts that alter resource allocation (including adding a new resource) will be purely incompatible. And so on. For anything other than GameData XML and SQL, a mod needs to replace the original file in its entirety, which by definition means that two mods can't both apply their changes to the original. The game will simply use one of the two, depending on load order, and ignore the other. If the ignored one did something important, you'll see problems right away, but if the differences are minor then it might be a while before something breaks.
If you have this sort of conflict, there's not really anything you can do about it except not try to use those mods together any more. It's theoretically possible to reconcile the two, but merging two disparate version of a core game file can be VERY complicated, and you're likely to break it badly in the process.
Now, it's also possible for two mods involving purely XML or SQL updates to be incompatible, especially if the mods add new elements (notably text keys). It's just far less likely, and much easier to fix.
Additionally, if possible, i'd like to know if there is some sort of loading log that I can access or implement when loading mods, so that I could not get 1/4th of a way through the game and have to ditch the save.
Your game has a "Logs" directory. Inside it, the game creates files called things like "Database.log", "Lua.log" and "xml.log". Those'll contain many of these issues, but not all of them. If two mods each attempt to modify the same UI window, for instance, there'll be no error message at all, because neither mod actually did anything incorrect. And the vanilla game generates a few errors and warnings in each of these logs, so you should see what they look like when no mods are loaded to know what's "normal" in a mod's errors.
Now, if those files aren't being created, it's because you never bothered to turn logging on. Go into your game's config.ini file and change EnableTuner, LoggingEnabled, and anything else with "log" in the name that looks even remotely useful, from a 0 to a 1.