They gladly received his word (faith), repented, were baptized, received remission of sins, and then received the gift of the Holy Ghost, in that order.
Now in Acts chapter 10 it must be considered that that chapter is about a time of transition in the church in which God had to prove that Gentiles could be saved in order to include them in the church (because there was much opposition coming from the circumcision group to Gentiles being saved). So God went beyond the usual boundaries to prove that they were to be included in the body. Note however that when Cornelius and friends received the Holy Spirit, Peter considered that they still needed to be baptized in the name of the Lord. "Can anyone forbid water, that these should not be baptized even as we?" This shows that Peter at the very least (on whose testimony Christ would build His church) believed that baptism was at the very least important.
re #106.