NCT TRACON Controller reported an airborne conflict which turned into an aircraft being below the MVA and having an aircraft equipment problem.

Narrative

Aircraft X is on a base turn heading 200 for the GPS approach into HWD. Aircraft Y is on a 050 heading to pass behind my traffic. I clear Aircraft X for the approach heading 260. Aircraft Y starts turning left into my traffic to apparently join V334. I tell Aircraft Y to turn right heading 090 immediately for traffic. The pilot hesitates and apparently has some kind of pitch malfunction which caused them to descend into a 040 ft. MVA. I issue the low altitude alert and tell them to climb immediately but they're all over the place climbing and descending above and below my assigned altitudes. They get in the way of the SJC departures climbing to 050 and almost got into the way of the SFO arrivals. I issue multiple headings and altitudes and they did not comply. I asked if they required any assistance and they say they're working something out. Once they're finally on a stable heading and altitude I ship them to Sunol.Since I have been working Mulford/Grove; I have never seen an aircraft just turn to join V334. Apparently; the SOP exit route for Toga is V334 or a 050 heading. I have only ever seen aircraft on 050 headings from Toga. I obviously had traffic at 050 inbound to HWD. The Toga Controller should've reached out and asked what I wanted or just vectored Aircraft Y. SOP exit routes do not trump our duty priority as air traffic controllers. You do not ship a deal to another controller. The SOP exit route needs to change to 050 headings only.