ett läge = a situation
läget = the situation
det läget = that situation
"Läge" can mean "position" as well, but here "situation" is a better translation.
For an ett-word, yes. For an en-word, it's "den + ~en", for example
den perdioden = that period
Isn't this an ambiguous sentence? "läget" has two meaning according to the dictionary: location and situation. Couldn't someone ask "what does one do in that location?", meaning in that place?