Translation:František has been living there for a year or two already.
"František is already living there ..." is not accepted. Is this an English idiom issue?
To my native AmE ear, "IS already living there for a year or two" sounds wrong. I'm fairly sure that "has been" is what's needed here, since it takes care of both his past and current "living."
I think it is not correct with the time interval, but I may be wrong. But it does exist on its own.
Using the present progressive here is possible in American English, but it's quite informal. "So Frank's living in this dump for two years already before he finds out the sink doesn't work."
'Franistek has already lived there for a year or two' should be an accepted answer.