Playing as the Empire, I'm not a big militarist (I usually rely on mercs and the Academy flagship to rush someone in early game, it works on Hard). The UC were my neighbours and they started to annoy me. I got enough money to get the Academy fleet (which has detection), I found the UCs home node, and got the popup about it. However, there was no option to invade on the node itself. Why? How was it possible for me to find the hidden system, but not invade it? What do I need to do to invade it?

Well, did you actually see the system itself though?
Because once you spot it, UC, especially AI UC does instant beacon teleport of said system somewhere else, unless of course you were somehow blocking it, or sieging it by being at war with UC.
Try reloading the autosave and blocking/sieging instantly. You probably might catch it while it is still there.

Actually, I had the same issue with the UC in another game, was playing Hissho, had the Spear. 

The problem was, indeed, that UC had used their beacon. I don't understand, though, how they were able to do it during MY turn. I mean, on my turn I use the spear to go to a nebula, I get the popup saying that it's the UC's home system, but they are already not there. That's very misleading. In the aforementioned Hissho game they actually beaconed to a neighbouring nebula, so I caught them.

There isn't such thing as YOUR turn, all turns in Amplitude games are simultenious. Turns only used to provide resources, build structures, research technologies, refresh cooldowns, movement points etc.

Ok, that's good to know. I thought it was a reactive implementation or something like that that would work without blocking you from doing your turn, but if that's intentional, that's a whole other thing...

Will there be an achievement for defeating the Umbral Choir?
Or having them relocate a certain amount of times?
Because I think that in itself would be an impressive feat :)

I mean they do pay a price for a relocation, so I d say it's still a win even if they are out of harms way.