Did the Gentiles in Acts 10 receive the Holy Spirit before repentance and baptism simply by believing?
Yes. In Acts 10 the Gentiles received the Holy Spirit while Peter was still speaking, before any call to repentance or water baptism. They simply believed the message about Jesus—what happened to Him and His resurrection—and the Spirit came upon them as evidence of their faith, reversing the Acts 2 order.
Les explains that Acts 2:38 gave the criteria for Israel: repent, be baptized in water, receive remission of sins, and then the gift of the Holy Spirit. But in Acts 10, the Gentile economy is totally reversed. While Peter was still speaking, before he said anything about repentance or baptism, the Gentiles believed the message concerning who Jesus was, what had happened to Him, and how He had risen from the dead. The Holy Spirit then came upon them as evidence of their faith. They believed without baptism and without repentance. Les notes this is Paul's Gospel, though it was only the first step of opening the door to Gentiles.
While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.
More Questions
What did Peter preach to the Gentiles before the Holy Spirit fell in Acts 10?
Peter's preaching to the Gentiles in Acts 10 was about who Jesus was, what had happened to Him, and how He had risen from the dead.
Had the Gentiles in Acts 10 been baptized before the Holy Spirit fell on them?
In Acts 10, the Gentile hearers had not been baptized and had not even heard anything of the Law, but the moment they believed Peter's message the Holy Spirit fell on them.