In Diplomacy Distant Stars was perhaps the closest thing we had to "en masse". But it has not been kept entirely up to date for Rebellion. There is no matchmaking support for mods, you can play them online over ICO but people who aren't playing the same mod cannot join your game, and there's no easy way to communicate this (save putting the mod name in the game name). However, since no one expects to see mod games randomly on ICO, you pretty much have to arrange to play them ahead of time (a few do have steam groups or other ways of keeping track who is interested).

The Halo mod just released the first playable version, and since it's Halo all of the fanboys are swarming over it (Maybe it does deserve it but it and the Mass Effect mod have been highly ranked on moddb for years, despite them not being playable).

Basically I would just find a mod you like and post to see if anyone is up for a game. This is my favorite list, otherwise just say what you're looking for and I'm sure people will give you recommendations.

Sins of a Galactic Empire has a decent MP following; we usually play pick up games (one person mass-messages the Steam group to see if others are willing). Over the past number of weeks we've played at least a dozen games.

Add Mass Effect/Star Wars/Halo playable factions. Some of them might be "total conversions", in which case the original factions are taken out, and likely change a lot of other things as well, but the main thing is the new playable factions.

I highlight which mods are which in this thread https://forums.sinsofasolarempire.com/423212.

You essentially have 3 types of mods.

Total Conversion - Aims to replace all assets from the original game with new ones to fit their universe. Sometimes a long road to fully replace all vanilla assets.

Expansion - Aims to add additional playable content to the original game. Could be new races/factions, but also could be extensive gameplay enhancements like E4x.

Graphics/Effects - Aims to improve the overall look and feel of the game. This type is typically included in TC or expansion mods. 

In theory yes. Any mod with a "Checksum" of zero can be used in multiplayer. I always use a music replacement mod is MP for example that just loads a different set of sound files for example. However, for both textures and sounds, they have to be set up in such a way to "Replace" what is already in the game. If I tried to add a new sound effect for example I couldn't use that in multiplayer, even though it doesn't really affect the gameplay balance at all.

I have only seen maybe 1 planet texture pack ever released, most planet mods add new planets, and thus cannot be used in multiplayer.

No it doesn't, at least not the Rebellion one I saw. It changes the ships to tell them to add new particles (Rebellion added new particles, so to get it to work with Bailknights the guy who updated it had to change every ship to use the old ones). You can try deleting the "GameInfo" and "Mesh" folders of any mod to get rid of the checksum, but they may not work as you think (when I tried it with Bailknights, I only noticed the explosions showed up).