Was Joshua Jewish

Yes. He was an Israelite of the descendants of Ephraim, son of Jacob.

Joshua had been Moses' right-hand man since his youth (Numbers 11:28), and his chief disciple; and was constantly found in the tent of Torah where Moses taught (Exodus 33:11), so as not to miss a single Word of Moses' teaching. He could be trusted to reach correct decisions (Numbers ch.13-14); and was successful in battle too (Exodus ch.17).
As Moses approached the time when he would have to hand over the helm of leadership, Joshua turned out to be the best choice to carry on, and God named him as Moses' successor (Numbers ch.27). It was Joshua who led the Israelites into Canaan (Joshua ch.1).