20 turns. This is defined in GalCiv3GlobalDefs.XML if you want to change it, but be aware that the changes will only take effect in new games; the line you'd want to look for is

<CannotSpeakToPlayerAfterDeclaringWarDuration>20</CannotSpeakToPlayerAfterDeclaringWarDuration>

To change the time period, change the highlighted number and save the file, either overwriting the base game file or as a copy in the mod directory. In the latter case, you will need to enable mods before this will take effect, you will need to have the file inside a \MyMod\Game directory inside the mods directory, and it will conflict with any other mod you have loaded that also modifies GalCiv3GlobalDefs as only the last GalCiv3GlobalDefs loaded will be in effect.

Not quite. Regular diplomacy with computer factions only requires a 15 turn wait.

Probably, but it's possible that they might have been nice and made it so that the game rules are pulled from the host rather than the local player. I'm a bit doubtful, but it's possible. I'm also not entirely certain that mods work with multiplayer, at least at present, as a lot of things are supposed to have been disabled for multiplayer for stability reasons.