I will try to answer your questions as best I can.
Chertio wrote:I have a siege going at Vicksburg (picture attached) : Grant is besieging the place, and I have three forces in the surrounding regions (Adams, Claiborne, Yazoo) which I put there to exert ZoC and prevent escape.
I also tried putting ironclads on the river exit (Joseph Run) but they kept getting beaten away by a CSA fleet.
Can the enemy escape directly from the fort by river transport or must they get out of the fort (and evade or defeat Grant) first?
Yes and Yes. They can leave by river transport since the port is not blockaded, but I believe they must evade Grant to do so.
Chertio wrote:Are my three surrounding forces achieving anything useful, or is Grant alone enough to keep the CSA trapped inside the fort?
Grant is enough to keep up the seige, but your other forces are keeping the military control in those other zones, whether that is useful or not is up to you.
Chertio wrote:How long do sieges last, and is there any way to judge how soon they might end? I have been getting messages about besieging and making breaches for a few turns, but this turn nothing (not even a "no effect" seige message). The depot in Vicksburg looks permanently empty, do I have to wait for the troops to starve?
The siege will last as long as it does, there are not set times. The breaches only let you know the status of the defenses and how effective the will be now if you assault the city. The siege may or may not end before the troops inside starve. Since you do not have the port blockaded, the city
can continue to recieve supplies at a reduced rate (50% I think, someone correct me if I am wrong).
Chertio wrote:Also I managed to foul up the turn succession last turn and may have reconstructed it wrongly, maybe that might account for CSA force apparently being stuck in Vicksburg (PBEM, my opponent tried to exit last turn and failed).
If your opponent drops his forces outside of the city there is a normal chance of battle like any other, but I believe if he losses, he will got back into the city. I believe the same is true if he tries to move out of the region with evasion, if he fails he goes back to the city.
Someone else may correct me, but I think this is how it is.