The campaign was interestin'. It starts of with a few sectors and gets larger.

As a dedicated sandbox player since the OS/2 days, I never even tried the GC1 for Windows campaign until well after GC2 came out, and when I did it CTD'd before I could finish it (Stardock support, as good as they usually are, could not help me with it).  I was also never able to finish the GC2 campaign because I sucked at battling the dread lords.

Is there a text description somewhere that explains the events of the campaigns, and how they affect the lore of the GC universe?

No, each mission is on a different map. There are a few exceptions to this, though. In those cases, you restart the map, but there will be changes, because of something that happened at the end of the last mission.

Overall, I'd say it's worth it play the campaigns. They provide interesting background information (in my opinion at least), and offer some unique challenges (more so in DA and TotA).

However, there are also a few bugs. One mission in the DL campaign can be unwinnable, because the map hasn't been updated (the win condition is set for a race that isn't on the map). The rest are less severe. Mostly some missing text or a planet hasn't been set to be the homeworld.

I fixed all of those bugs (plus a lot more in the tech trees, planetary improvements, etc.) in my mod, so you might want to give it a try, if you decide to play the campaigns.

I tried campaign once. But I t was boring, compared to the sandbox. So I only read XML files for the story.

That was because each mission in the campaign was essentially a separate scenario. However, it wasn't as bad in the DA and TotA campaign, because you didn't start over from scratch completely in each mission.

I think it could be interesting to have the campaign from 2 different approaches:

Homeworld style: The stuff you do in one mission affects the next, more population means more colony-ships in the next area, techs are limited but the ones you research are saved and fully researched for the next mission, possibly even ships that have levelled up could be transferred to give players a starting fleet and an opportunity to keep the action up from the go.

Supreme Commander Style: The whole campaign is played on one map, but from the start you go from one system, and once you complete objectives the map expands to reveal new opponents and goals.

For anyone who hasn't or doesn't want to play through the campaign there is a great summary/timeline of the story available here:

https://www.galciv3.com/databanks/our-story

It covers what happened before the campaign started (History of the Drengins and Arceans, the Stargates, how the Terrans took their place among the stars), what happens during the campaigns and what situation is at the end of GalCiv2.

Sorry for the resurrect, but the topic I want to discuss is the same as this.

I have played a couple of campaign missions, now all on Suicidal difficulty. But in order to win I have to exploit what I know about the AI. Things like the AI attacking and almost constantly following the closest armed space ship no matter if the AI ship will never catch the armed ship. Another exploit of leaving one of my planets defenseless so that the AI would sends unescorted Troop Transports, which I intercept.

igncom1 & The Sisko have some really great ideas.

I have never heard or played Imperium Galactica, but it sounds like they fixed the issue that Supreme Commander had. In Supreme Commander you may often lose from the next phase if you follow the "hurry and finish the objective" advice from superiors. So I always prepared before moving into the next phase. In Supreme Commander: Forged Alliance, this method broke the campaign, as I was ready send multiple super units to finish the objective of all other phases. But if you don't know anything that is upcoming from the next phase, how can you proceed unprepared?

In Galactic Civilizations II campaign it is supposed to be that the player is controlling a regional force from the rest of the faction(eg "Bradley", "Jenna").
Galactic Civilizations III could go the route of having a active/semi-active large galaxy, but with the player only controlling a part of his faction with restricted movement and visibility. And after completing , open up the next phase. The active galaxy is there to prevent the player from intentionally stalling on completing the objective in order to prepare for the next phase.

Tried the campaign when I first got GC2.  Ran into a level early that the Dread Lords just killed me every time.  Gave up.  Tried it again some time later, thinking I was better at the game.  Got killed again.  Repeatedly.  Haven't gone back.  I am not fond of campaigns in the first place.   This one didn't change my mind. 

I will keep playing in the sandbox and being very happy with it.  Got a shovel I could borrow? 

I've played GS2 for year but did not know it had a campign until now.  I don't usally like campaigns in 4x games anyway since I perfer the sandbox part of the games.