It actually means content, satisfied. Contentivly happy, satisfactorily happy. However those that say it, I find are usually lying. Trying to get the point accross that you have lost something or can't sell them something.
Either. It depends partly on how it's said and partly on what follows. "I'm quite happy, but still concerned about..." would suggest "slightly," but "I'm quite happy: it's my birthday!" would suggest "very."
In writing without any other context I'd tend to assume "slightly" because it seems to be the standard usage of the word. Even when someone says "Careful, it's quite hot," they don't typically mean very hot. It's "Don't get surprised and drop it" rather than "Seriously, get some oven gloves."
Depends on context. It’s one of those modifiers which is not required in written English, but is an instinctive reply to certain spoken situations to put more or less emphasis on a particular situation. Without judging speaker tone it’s hard to say whether they mean ‘very’ or ‘slightly.’ I went with slightly, but in hindsight it possibly has more relevance to a greater quantity. “That’s quite enough” would imply one has done more than needed. For example. Same for “Not quite there yet.” It could also be deemed as slightly depending on what the speaker believes it quantifies or what the receiver deems also.
Oh, English. Why must you be a beautiful, confusing mess of a language? I love it.