1) Please explain what sign to an unbelieving Jew even means. Not following you on that one.
He said it was a sign to the unbeliever and I think of when Jesus sent his disciples to heal and to say that the Kingdom of God has come nigh unto thee. Some would still not believe but it would be more tolerable for Sodom and Gomorrah in the day of judgment than for that town. Tongues was prophesied but for all that they will not believe. So it is a sign to the unbeliever as a testimony that the prophesy proved that what God said came to pass but they still did not believe and will have no excuse on the day of judgment. Then he gives and example of prophesy and says an UNBELIEVER comes into the assembly and by the manifestation of the gift of prophesy has the secrets of his heart exposed and BELIEVES that God is in you of a truth. Now he was an UNBELIEVER and yet prophesy was used to cause him to BELIEVE. When people say that prophesy is for believers and tongues are for unbelievers they usually are confused as to what Paul was saying and as a result they say things that make no sense. So what do you mean Tongues are a sign for the unbelieving Jew?
Exodus 4, 1 Kings 18:36-39 and John 10 all explained what signs to unbelieving Jews meant.
I will just quote the last passage
24 Then
the Jews surrounded Him and said to Him,
“How long do You keep us in [d]doubt? If You are the Christ, tell us plainly.”
25 Jesus answered them, “I told you, and you do not believe.
The works that I do in My Father’s name, they bear witness of Me.
31 Then the Jews took up stones again to stone Him. 32 Jesus answered them,
“Many good works I have shown you from My Father. For which of those works do you stone Me?”
33 The Jews answered Him, saying, “For a good work we do not stone You, but for blasphemy, and because You, being a Man, make Yourself God.”
34 Jesus answered them, “Is it not written in your law, ‘I said, “You are gods” ’? 35 If He called them gods, to whom the word of God came (and the Scripture cannot be broken), 36 do you say of Him whom the Father sanctified and sent into the world, ‘You are blaspheming,’ because I said, ‘I am the Son of God’? 3
7 If I do not do the works of My Father, do not believe Me; 38 but if I do, though you do not believe Me, believe the works, that you may know and [f]believe that the Father is in Me, and I in Him.”
That is what signs for unbelieving Jews meant, from scripture. A Jew requires signs, and is unwilling to believe you by faith. You show him signs that you are sent by God, so that they can believe those signs, and believe you are sent by God.