"My cat is usually at home."
Translation:Katten min er vanligvis hjemme.
"Vanlig" means normal, so I guess it could be translated to "normally", while the other, "pleier", would be more like usually. I typed "som regel", which seems to be wrong, though it sounded right to me (I'm Norwegian, just running through this stuff). As far as I can think, they can be used interchangeably, yes.
Et hjem = a home Hjemme = at home
"Hjem" alone can refer to just "home", such as "Dette er mitt hjem" ("This is my home.") "Jeg kommer hjem." ("I'm coming home."), and it can also be used a bit differently such as "Jeg vil hjem." ("I want (to go) home.").
"Hjemme" is usually when referring to the location, as in "Jeg er hjemme." ("I am home.") "Jeg er snart hjemme." ("I am soon home.")
At the moment, "jeg er hjemme i leiligheten min" ("I am home in my apartment."
I guess this might be something that'll come along as time goes by, even I had to think, as it kinda falls naturally for me. I'm also no language expert, though I did me best, and if there's anything else, feel free to ask~
They are controlled by the verb type. If the verb is stative (er hjemme, er inne), the 'e' stays. If the verb is dynamic (kommer hjem, går inn), there is no 'e'.
I always imagine a word chilling on top of the car roof with 'e' loosely attached to it. When the car is in motion, the 'e' falls off. But when the car is not moving, it stays with the word. :)