Yes, but it also means "our", and agrees with noun. vår katt, våra katter, vårt barn.
The translation in the sentence is clearly the wrong sense, however.
I think "tycker om" is more passionate than "gillar", especially when saying it to/about other people.
Not really, vår is for en gender words, vårt is for ett gender words, and våra is for plural.
In English, you use ours when it is standalone at the end of a sentence or so, like It's our cat but The cat is ours, but we don't have a counterpart to that, it's the same in both cases: Det är vår katt and Katten är vår.
Really, Duo couldn't tell that 'out cat' was a typo of 'our cat'? Come on, T and R are immediately next to each other on the keyboard. 'Out cat' doesn't even make sense in this context.
The spelling rules are not decided by us course creators, but by Duo. Their principle is that you can have one letter wrong so long as it does not turn into another word. If you want them to change their algorithm, feel free to bring it up in the general forums. We can't do anything about it either way.
I am not sure what you meant by "vur" (not a standard pronunciation scheme as far as I am aware) but in IPA it's /voːr/