After reaching a decision. How did the elders send it sent to Antioch?

The sent a letter and Judas called Barsabbas and also Silas. Presumably Judas and Silas could vouch for the letter.

Source:

Acts 15:22-29

James was quoting the Prophet Amos (Amos 9:11-14) to support his judgment .

Source:

Acts 15:16-17

So then James the leader of the Jerusalem church rendered his Judgment by quoting a prophet. Who was the main prophet he quoted?

According to Peter, how would God purify the hearts of the gentiles?

God will purify the hearts of the Gentiles with Faith. Just as he does with the Jews.

Source:

Acts 15:9

What countries did the delegation pass through on the way to Jerusalem?

The delegation passed through Phenice(Phoenicia) also Samaria. Most of Phenice is part of Lebanon today, also Samaria is part of Israel.

Source:

Acts 15:3

In brief, what actions were the immediate cause of the Jerusalem conference?

Jews went to Antioch telling the Gentile believers they must be circumcised and also keep the law of Moses to be saved. Barnabas also Paul disagreed.

Source:

Acts 15:1-2