I'm about to start getting into the civil war portion of the questline, and I already sided with the Stormcloaks. Does winning the war result in Stormcloak soldiers occupying EVERY town, city, and fort in Skyrim? Does this mean that you can no longer find Imperial Soldiers on the map?

If so, I'm sort of torn about the idea of winning. I like running into imperial soldiers randomly while they escort stormcloak prisoners. I like attacking their forts and camps. To what degree is the imperial presence removed?

The Stormcloaks will have control of all towns you'll still find patrols etc. from imperials.

You will still encounter imperial camps. There's a bug, though, and the generals in the camps are still marked essential. If you're on PC, there's a mod for that.

Not sure about the patrols.

The Stormcloak soldiers will occupy all forts and all former pro-Empire cities. I think I have encountered random patrols of the losing side post-civil war, but i cannot verify that. Celan is right in the camps still being occupied by Imperial soldiers.

Celan nailed it. You will still always find camps of the enemy but the bug prevents you from killing the commander.