>Were there any passenger trains over the bridge?

There were at one time, but that was before the late 1920s/early 1930s -- there was service SF-Sac via Redwood, Newark, Niles, Altamont, Tracy, Lathrop, Brighton, Elvas, and finally terminating in Sac. I forget the daily frequency.

>How many trains traveled this route VS going to San Jose?

When the Mtz bridge was built, the plan was to run trains from RSV to SF down the Cal-P via the Mtz bridge, at the west end of the bridge there was to be an interlocking plant to rout the trains out the Mococo line to Avon, down the San Ramon branch Avon to East Pleasanton (or Radum, forget the name of the station where the San Ramon branch joined the Tracy line), then East Pleasanton-NWK-Redw Jct on the Tracy line, then Redwood-SF (Bayshore or MB) on the Coast. This was all done in by the depression, and the never-used interlocking tower at built at the [geographical] south end of the Mtz bridge to control the planned junction interlocking wasn't torn down until in the early 1960s when the first Benicia bridge was built. Until the main lines were rearranged when the CTC was installed Mococo-Port Chicago, the storage track with all the tank cars was called "the Tower" because of this interlocking tower. It was right where the first highway overpass was built.

As for which route had more trains, that depends on the era. Most of the trains which went Oak East to SJ either terminated there or continued down the coast (and vice versa). There was essentially no connecting track to go from SJ to NWK via Redwood Jct. Until the connection between the Niles sub and Cal-P was built in the early 1906s in W oak, right where BARTD goes over the main lines on its way to the east end of the transbay tube, there was literally no way to go from the Cal-P to the Niles sub without a lot of sawing back and forth.