Isn't this slang for penis? - The answer is yes.
Doesn't this mean "I am a dick"? - This sentence absolutely doesn't mean that.
Does that have to be "cái chim"? - No, we always use con for chim.
Although the word chim also means penis (slang), we recommend you not to use it with this meaning in this course and in real life as well. If not, you'll be impolite.