The correct answers are The ticket in the evidence bag is mine. and The manager gave me free popcorn.

Mine correctly shows ownership without needing a noun after it.

Me is the correct object pronoun to receive the action of the verb "gave".

"My brother and me..." is incorrect because you need the subject pronoun I when you are doing the action ("My brother and I went...").

"...to my" is incorrect because my must be followed by a noun (like "my friend"). Here, it should be the object pronoun me.