Were proselytes considered "Jews"?  For instance, would a proselyte have been welcome in the temple or synagogue? 

A person who converted to Judaism was considered equal to those born under the Jewish law. Yes, they were welcomed into the synagogue. When Paul was teaching in the synagogue, there were proselytes there. "Now when the synagogue broke up, many of the Jews and of the devout proselytes followed Paul and Barnabas; who, speaking to them, urged them to continue in the grace of God" (Acts 13:43). They were also mention being present at the feast of Pentecost (Acts 2:10).

See also:

Questions and Answers regarding Judaism