bambino, "search me" can also mean "beats me/I don't know", if it's used in the same context that "beats me" to mean "I don't know".

No, they are two different things. "search me" is another way of saying of "Check Me" " Beats me" is another saying of " I don't know."

i'm not sure because search is 搜索. maybe it can be used or it is similar to search for me, such as search me the meaning of something. But it sounds a bit improper. xD it can be similar to i don't know because if you do not know something, you may search for it or ask someone to search it for you.

I've never heard of the phrase "search me" and I have no idea how to use it in proper context. It must be a regional phrase or British English.

I've actually never heard "search me" unless it was to prove that they didnt take something! like if somoene is accusing you of taking something that you did not take, you can say "search me" "I dont know" is used when you have no known answer for something ^^;; hope I helped