Click the start button on the telescope. I've found that it usually stops instead of starting another search - this might be related to credits in the bank?

I totally didn't see that start button. I thought "stopped" meant there was nothing left to do. If there are more than 20 sectors, than idk how many. It is really hard for me to find any details about the gameplay of this game...

Still, I'd like to know, are the artifacts in "sectors beyond the first 20", or do I need to do something to find them in sectors I already found, like visiting certain places?

If there's an artifact in a sector you'll see the Unknown Signal indicator on your HUD, no matter how far away you are in that sector.