... let's see
1. First thing, an army's starting movement points is equal to that of its slowest unit, + any bonus to movement from general. Every tile it moves to takes away from its movement points. Certain terrain takes away more movement points than others (Wilderness, Desert; trekking from Antioch->Baghdad, no roads, it is faster to walk along the greenish tiles on the side of the river). Roads lessen the amount of points lost.
Now, you say your army marches from the city to a fort, and i''m guessing the fort you have the problem with is off the road. If you start from the city the green arrow path you'll take will try to take advantage of roads as much as possible to conserve as much movement before leaving the road, maximizing movement. But from the fort to the city, by the time you reach the road you have depleted much more movements points, before you even reach the road, that there isn't enough to reach the city, or you don't have the general with you. Note that generals may have a penalty to road movement and if there is more than 1, it could actually slow down your army, i.e. 'Slothful', or 'Easy on the men' (SS mod). (if this is confusing, just say so, and either I or someone else will try to explain it better)

2. I believe this has got to do with Population, the numbers and the factors affecting this. You can get an idea from the city scroll details (the one with icons for trade, public order, p growth). The higher the negative factors to public growth (squalor, taxes) compared to the positive effects (farms, governor chiv points, bonus from own hall) the slower the city grows. At worst, the city stops growing or the city will lose population next turn.
Anyway, a place that has very low growth or a bad population (+) : population (-) ratio has a higher chance of catching a plague. Also, plagues are more likely to happen to large/huge cities compared to towns, possibly regardless of population growth, since I never saw a plague happen to a large town (despite having a growth rate of +0.5-0%)

3. I have litle experience with elephants, ditto facing them on the field. All i could think of is a defensive battle, choke point (bridge, city street, gateway), rotting cows-flinging trebuchets +archers or other siege support. Elephants have high morale and you need to lower it fast before they reach you line, it helps if their comrades aound them rout fast. Bridges and chokepoints works best. Rotten cows, just as they start crossing the bridge/chokepoint, will lower morale considerably of all units that cross it. After 1 or 2 flings you switch the trebs to fire ammo. Fire artillery + arrows (fire or not, a mix will do i.e. weak archers use fire arrows, better archers use normal arrows to cause more casualties. Don't forget to target where the enemies are tightly packed) will cause casualties and and even lower morale (at least to non-elephant units also trying to cross) before they clash with your defensive line (decent spears and heavy infantry). Once a rout has started (hopefully not yours) the elephants should follow suit (unless they've already run amok among enemy lines). That's all I can say as i've not had any success fighting them on the field.

4. No clue, sorry