A couple of Questions

I was asked the following:

1 Ne. 10: 11 And it came to pass after my father had spoken these words he spake unto my brethren concerning the gospel which should be preached among the Jews, and also concerning the dwindling of the Jews in unbelief. And after they had slain the Messiah, who should come, and after he had been slain he should rise from the dead, and should make himself manifest, by the Holy Ghost, unto the Gentiles."

Is this talking about a physical manifestation to the Gentiles? Does Christ show Himself to others physically by the power of the Holy Ghost? Or is this to the Gentiles' hearts and minds before the Restoration?

This is speaking about the immediate post-resurrection ministry of the Lord. At that time He visited only with the tribes of Israel in their scattered condition. He did not go among the gentiles. Nephi explained that in the last days ministry of the Lord at that time, He would appear to the gentiles "in very deed." (1 Ne. 14: 1.) This is why the Lord appeared to Joseph Smith (JS-H 1: 17-19) and Oliver Cowdery (D&C 110), and to Sidney Rigdon (D&C 76), and to others.

_______________________________________________________________

In Mosiah16:1 and Alma 13:21, it says the phrase "he stretched forth his hand." What does that mean? Raising it to the square? Using the priesthood to testify of what he is about to teach? A little help here would be wonderful.

Read Mosiah 15: 31 to understand 16: 1. He is demonstrating the Lord's action, thereby affirming he is His messenger. He had been given the sign to testify, and used it as his sign that he was a true messenger.
In Alma 13, the prophet concludes his testimony of Melchizedek by using a sign to evidence his authority. He used this sign because he was authorized to do so, and knew what it meant as he did.  Although those who were there may not have understood, it was a sign he was a true messenger.

We cannot be saved in ignorance.

Once the key of knowledge is lost, mankind is lost and cannot be saved until that key is returned.  Prophets sent with messages who testify to an ignorant people use signs that the Lord recognizes and authorizes, but they may not be noticed or understood by those who hear the message.  Nevertheless, the testimony becomes binding when the Lord's seal is put upon it. This often involves a required sign to be given, or in other words, for hands to be stretched forth.