Hiya. Campaigner is a pretty good solution for a large part of what you're asking, though there are only two factions (but any number of players) and no real supply system to speak of. There were lots of great additions that we wanted to put in (a more complete supply/logistics model) that just didn't make it.
Unfortunately, not enough interest in Campaigner meant a lot of great features just aren't in it.

But, I'll answer what we DO have:
Custom Map Generation: via FlexMap, Campaigner uses point-based movement, so no hexes or squares.
Multiple Factions: There are two opposed factions, any number of players per faction though.
Armies: As many units as you allow can wage battles across the map.
Political Alliances: No, I wish it was.

LAN: Not so much unless copying the file counts
PBEM: Absolutely
Fog of War: Wonderful FoW system if you ask me. Uses the "size" of the unit as well as terrain and its mapped THROUGHOUT the turn, thus an enemy can "cross" the line of sight of another unit and disappear.
Supply: Yes and no. Custom resource generation is in Campaigner, as well as custom rules. The rule/resource system is quite robust, but doesn't automatically require resources for units. That bit just didn't make it into Campaigner.
Battle Resolution: Yup. Done on the tabletop and you tell Campaigner what was killed and who won.
Strategic Objectives: We got 'em, whether based on area objectives, destroying installations, killing units or gather a specific resource (or any combination), Campaigner has an excellent victory conditions system.
Rulesets: Campaigner doesn't care one iota what game system you're using, doesn't care what era it is either.
The two major holes in Campaigner (if I may be so bold) are a lack of a fatigue/supply system. While we do have resources for custom rule use, there isn't anything for automatic fatigue or supply consumption.
The other, is the simplified diametrically opposed factions. A player is either with THEM or THOSE GUYS (Red or Blue or whatever you rename them to).
If there was enough interest in Campaigner, we'd see to it that multiple factions and fatigue/supply would be included. Support for Campaigner HAS NOT ceased though. If there is a bug, it'll be fixed, its just that new features would be very slow in coming out.
If you need a specific sample campaign, let me know. I'd be happy to throw together a demo campaign that you can play with the free version of Campaigner.