Question for natives: can tenacious mean something bad?
As I understood tenacious means "persistent in maintaining" like "tenacious defense attorney". But it also has a meaning "tending to adhere or cling especially to another substance" like "tenacious weed on the lawn that we can not can a rid of".
Are these meanings applicable to a person in a "bad" or sarcastic way? For example, "our new colleague is so tenacious in her efforts to make friends with us that she follows us everywhere?" or "he is extremely tenacious; I can't get a rid of him."
Are all the examples that I made correct?

Question

Question for natives: can tenacious mean something bad?
As I understood tenacious means "persistent in maintaining" like "tenacious defense attorney". But it also has a meaning "tending to adhere or cling especially to another substance" like "tenacious weed on the lawn that we can not can a rid of". 
Are these meanings applicable to a person in a "bad" or sarcastic way? For example, "our new colleague is so tenacious in her efforts to make friends with us that she follows us everywhere?" or "he is extremely tenacious; I can't get a rid of him."
Are all the examples that I made correct?

Answer

Yes, used in the way you presented it, tenacious can have a negative meaning.

Answer

Your examples are good but "tenacity" in a person is generally positive thing. So it conveys a rather more positive tone than sounding like a complaint.

He's harassing me. (very negative) 
He's bothering me. (negative)
He's pursuing me tenaciously. (suggests that the speaker respects his efforts)

Answer

All of your examples are right. The word "tenacious" itself has no positive or negative connotation. As you have shown, it is only the context around it that decides if it is positive or negative.