Translation:Our boss is annoying.
I feel this sentence has several issues. 麻烦 is closer to "troublesome" than "annoying", and "我们的老板给我们(添)很多麻烦" (Our boss makes a lot of trouble for us/gives us a lot of trouble) is much more natural of a sentence than directly using 麻烦 to describe someone.
Yes, when 麻烦 is used to describe a person, it means the person is a troublemaker. To say the person is “annoying”, we should say (很)烦人 or (很)烦, e.g. 我们的老板很烦人.
烦人 = annoy + people = to be annoying to people
Yes, so I take it as a very colloquial sentence, since, although 烦人 is the accurate phrase, I hear many people describing someone as 麻烦.
Right, I'd say both are possible. 麻烦 is troublesome, e.g. the person often comes to you with new demands. There's a more colloquial term with the same meaning in Beijing dialect: 事儿 ("我老板很事儿"). 烦人 is annoying, e.g. the person keeps talking to you about something you don't want to hear, and you find that very disturbing.