Difference between 发愁，着急，担心，关心，操心，烦恼
发愁 - The things to worry are like huge fruits hanging on their stem. It burdens.

着急 - So worried to urge oneself to "do something!" (at all)

担心 - Concerned about something, say a friend's health. The heart feels like hanging in the air.

关心 - Generally care about something, may or may not to an extent of concerning. (Btw, I so agree with your definition for this one)

操心 - Devote oneself into something, put a lot of efforts or thinking's into, because one cares about it or is concerned about it. For example, a parent may be so worried about the child's study, that they take a lot of efforts to research which school is better, which teacher is better, which major is safer, etc.

烦恼 - It can be both a word or a noun, either "to worry about" or "worry". But the degree of worry can be a bit intense. 烦 means annoyance and 恼 means anger. They together lean more towards annoyance, or generally anything that's mentally disturbing or causes discomfort. Or as a verb, it means staying full of those annoying emotions.