In the time of the Judges this Prophetess prophesied under a palm tree?

This is Deborah who liberated Israel with Barak.

Source:

Judges 4:4-6

The Bible talks abut three prophets who served King David. They are Samuel, Nathan and who?

The third prophet is the seer Gad. He served David for many years.

Source:

1^st Samuel 22:5 and others

Only one book of the New Testament is considered prophecy. What is the book and who wrote it?

The book is the book of Revelation and the author is the Apostle John.

Source:

Revelation

This prophet was also the last Judge of Israel. Who is it?

This is Samuel last Judge of Israel and also a prophet of God.

Source:

1^st Samuel 3:20

Who did God say would be Moses prophet?

His brother Aaron would be Moses prophet to Pharaoh and later Moses high priest.

Source:

Exodus 7:1