When did the R-77 active-radar-homing Air-to-Air missile become
operational in Soviet service? Which fighter (the MiG-29 or Su-27) got
the operational capability to use it first?

In 1993 with the MiG-29.

Which of the two aircraft--the MiG-29A or the Su-27 "Flanker-B"--was
the first to enter operational service? I see service entry dates as
either 1984 or 1985 for both of these aircraft depending on sources.

Both the MiG-29 and Su-27 came off the production line in 1982. The
MiG-29 first unit was designated in late 1983 at Kubinka AB and
frontal aviation recieved them in 1984.
The Su-27, having repeated problems with its avionics, was delayed
entry until 1985.

Rob