Are there any current Protestant denominations which do not believe that Jesus was born divine (i.e. that Jesus was God at birth)? Even if they consider Christ's spirit as divine, are there any that believe that Christ's spirit united with Jesus at some point after his birth?
The doctrine you are describing is called Adoptionism. It is one of many Nontrinitarian doctrines. Nontrinitarianism is a catch-all category for all forms of Christology which do not agree with orthodox Trinitarian doctrine. Although some forms of Nontrinitarianism are adhered to by some mainstream denominations today, such as LDS and Jehova's Witnesses, adoptionism is not. There are no modern mainstream denominations that teach adoptionism.
The particular form of adoptionism which you mentioned (in which the divine Christ was united with Jesus at some point after his birth) was first popularized by Theodotus of Byzantium during the second century.