2. Is the "doomstack" phenomenon really as bad as people say it is? I only have experience with the 1 unit per tile mechanic of Civ V. And if it really is so bad, is there a mod that fixes it (perhaps also introducing a 1 unit per tile mechanic, which I imagine isn't too hard to program)?
You wouldn't want to play it if it did exist. Its important to understand that "stacks of doom" are an essential mechanic of the game. Its the way AIs (or you) win conquest victories. It wouldn't play right without them. Its not something that is broken.
To explain:
There are several strategics aspects to Civ IV. Military tactics is only one aspect, and it
cannot win you the game just by itself. Stacks of doom serve the essential function of being a "Lose condition." If you run into a stack of doom that you can't deal with, you lose. It's that simple.
Anyway, Civ IV has several strategic elements:
1) Overall plan and strategy -- Determine what the best strategy is for winning the current game, including contingency plans if your plan fails. Determine whether you want to win by culture, space, conquest/domination, or diplomatically, and what units/wonders/buildings/technologies you will need to achieve this type of win, and how to deal with the AIs and prevent them from winning.
2) Empire building -- Build the most productive empire possible. This requires min/maxing, especially on higher difficulties where the AI has built in handicap advantages.
3) The tech race -- Tech as fast as you can towards the technologies you need to win the game according to your chosen strategy. This involves focusing your empire on commerce and on strategic use of great people to "bulb" the correct technologies, as well as clever tech trading with the AIs. It can also involve espionage to steal technologies from the AIs, if you chose to focus on espionage.
4) The diplomatic game. In civ IV, the AIs roleplay as their chosen leader. You have to know what each AI wants, and keep good relations with them, or they might attack you. They all have their own personalities. If you can get them to friendly, they won't declare war on you, but it is difficult to maintain friendly relations with everyone, so you need a plan to deal with the AIs who are not friendly or at least pleased. You may be asked to join a war, and you'll get a diplomatic penalty if you don't, so choose your sides carefully. Specific AIs will never declare war on you when you are pleased with them, and some are bigger warmongers than others, so you have to have knowledge of what each AI wants and plan your strategy accordingly.
5) Finally -- the military tactics game. If you get into a war, you need to make better use of your units than the AI does. Good military tactics will allow you to win wars even at a production disadvantage and a unit disadvantage. However, this can only help you so much... which is the point of stacks of doom...
Stacks of doom are what happens when you completely fail at every strategic aspect of the game other than military tactics, to an extent that military tactics are not going to ever bail you out. Its just going through the motions of how you lose. Its not the only way the game does this: if you fall too far behind, an out of control AI might win by culture or launch a spaceship rather than going for a conquest victory.
Basically, if you see a stack of doom headed your way, think about how you could have done better this game, and go ahead and restart, since you've lost! If you had done better in the tech race, maybe you could have nuked that stack of doom. If you had focused on better empire building and production, maybe you could have conquered the world 300 years ago, or at least substantially weakened the AI that is now kicking your ass. If you had done better diplomatically and befriended this AI, that stack of doom might be headed for someone else. Maybe you could have already won a cultural victory if you had min/maxed your culture better. There's any number of things you could have done to avoid this problem. So if you see a stack of doom at some point (most new players will encounter this eventually), don't blame the game. Its not the game's fault you lost!