"whether" in english is almost always followed by "or not". the phrasing of "Do you know whether the cat is home?" sounds off to me. I will think of om as "if" not "whether" in this instance (yes i know, hvis= if. its just to help me remember and make sense of the phrase.)
there are other opinions about it. When 'whether' is used to refer to a choice between two possibilities, 'or not' is redundant. There is a school of thought which says (especially when writing) that redundant words should be left out. 'Or not' is only necessary if you're describing something that will happen in either case: I will eat dinner whether you are home or not.
Among the people that I'm normally around it's very normal to use "whether" without "or not". But I definitely would affirm that there is a gravity towards adding "or not" in many cases, and the reason for this, as far as I can tell, stems from the fact that "whether" (like both, either and neither) belongs to a mostly vanished dual grammatical number (as opposed to singular or plural number), which means that the word "whether" requires two options to contrapose. So, in "Do you know whether she's coming or not?", the "or not" satisfies the duality of "whether". Similarly, something like "I don't know whether he prefers red or white wine" has the duality built in with the two options. So in cases like, "I don't know whether he's coming", where the dual options aren't explicitly stated, the duality is still implicit.
Still a relative beginner here, but as I understand it, 'vet' is about knowing something, whereas 'kjenner' is about knowing someone.
I'm editing this now (a year later!): 'vet' is about knowing facts, or knowing something for certain. 'Kjenner' is more like knowing about or understanding. So you would use the latter to talk about knowing a person or a place, for example, and the former to say you know the person's name or where the place is, etc. Eg: Jeg kjenner den mannen. Jeg vet han bor i Oslo.