No, you can't. They have completely different meanings. In both your examples, the phrase you need is 'all the time'. "Did you just stay home all THE time playing with your cat?" This means 'the whole time' in the sense of 'the whole afternoon' or 'the whole day', for example. or 'My son asks me all the time...' , in the sense of 'constantly'. You cannot use 'all time' in either of the above situations. 'All time' is used only in set phrases such as 'What is the best movie of all time?'. This contrasts with ' of the 1990s' or '...of the past 50 years'. It means taking into account the entirety of the period of history in question.

"Did you just stay home all the time playing with your cat?" Could be written as: - "Did you just stay home and play with your cat the whole time?" or - "Did you just spend all your time at home playing with your cat? The next one is a bit different, but uses 'all the time.' - Do you just stay home all the time to play with your cat?. The second one is: My son asks me all the time, "Mom, do you love me more or do you love your mom more?"