We would normally say 'I have doubts ..' in the plural, not the singular, and no, they don't have exactly the same meaning. For example: 1. I have doubts about whether John is capable of doing this job. 2. I doubt whether John is capable of doing this job. If you say 'I have doubts..' , you think that perhaps John isn't capable. There is maybe a 5 out 10 chance that he won't be able to do the job. If you say 'I doubt whether', this is quite a bit stronger. It could be seen as a polite way of saying that, in your view, John is not capable of doing this job. It suggests that there is a 9 out 10 chance that he won't be able to do it.

It's 'I've got doubts' not 'I've got doubt'. People also say 'I have doubts.'

No, they're the same. I doubt it is a bit less formal, although when speaking to someone and using the word "doubt", that's usually not a word they want to hear.