Translation:I do not remember if he was cleaning the apartment today.
Even if people could say that, we'd prefer to keep to the literal translation where "mieszkanie" is an apartment, not a house.
Well, they are a separate word, but I think even people that live in actual apartments would say "clean the house" instead of "clean the apartment", so I guess it should be accepted. Technically even "I do not remember if he had been cleaning the house today" could work.
how come there is no "on" here, shouldn't it be included by default (esp when we have 2 different people here)
No, the sentence is perfectly fine. Don't forget that you have context in real life. The subject of the second clause will almost surely be known to you. If you were talking about Adam, this means that "I don't remember if Adam was cleaning the apartment today". "on" works, but in 99% of real-life situations it will be totally redundant and actually strange.
In previous lessons it has been stated that possession is implied. Can it also be assumed here? As in cleaning "his" apartment today should also be an accepted answer? Currently (30nov2018) it is not.