Firstly, Triggers, how do I use them? I've tried openning the options dropdown menus for the data entry fields in the trigger window, but they all show up completely blank. Or is this part not developed yet?
No one knows. 
I've actually talked to guys at Stardock about this. Apparently the Galaxy Forge new features for Rebellion, including Triggers, were mainly done by an intern. He has since gone back to school of course, and with him went any knowledge of what triggers, if any, exist and could be used.
So they are definitely not in the Galaxy Forge, but that doesn't mean they don't exist at all. All .Galaxy files are technically text files, so if we knew the right syntax for triggers in theory we could add them. But sadly, no one has any idea what they are or what the syntax is. So yeah, you're probably better off labeling this as an unfinished feature.
Also, I heard players can share their maps (from galaxyforge) with the community if they wish. How do I go about doing the same?
If you want a Stardock solutions (I.e. you can use the account you are posting), there is a spot for them on Wincustomize. Or some people put them on as an addon on Rebellion's Moddb page.
How would they work in relation to mods? I don't yet have any personally (Mods that is), but many others do and I might want a few later on myself, would maps that I make in the galaxyforge still be playable for me if I have mods? and assuming they would be, would they also be playable for other players that have their own mods (even if I had no mods myself and visa versa)? Hmm, simplified, does a player have to have the same mods as me in order to play a map I create, or If I have no mods myself would that prevent other players from being able to play on them?
Many mods do work fine with maps made for Vanilla Sins. However, if you make a map for a specific mod that uses the unique planets/ships etc. in that mod, then yes, other players do have to have that same mod enabled. That's why you'll see some maps that have "SoGE, SoA2, Interregnum" etc. in the title, to indicate what mod the map is for.
Another question you should have asked here is DLCs. If you make a map using DLC specific features like Oceanic Planets or Ship Graveyards, you must have the DLC the planet came from. If you want to let all players use the map, try to use random planet lists when possible, rather than specific planet types. This lets players with the DLCs get a chance to have their extra planets on a map, while players without the DLCs can still play it. You should indicate which DLCs are required if you do make a map involving DLC planets.
