Should I connect the bare ground wire from the main panel to the grounding bar in the subpanel?
1) Replacing the subpanel is a very good idea -- FPE Stab-Loks are absolute junk that have a well-known propensity for failing to do their job when called upon, and also are susceptible to faults in the busbar-to-breaker contacts (high resistance/overheating) that are documented to cause the symptoms you're seeing -- this phenomenon can develop into a full-fledged electrical fire!

If a complete replacement of the subpanel is impractical for whatever reason, there is also the option of gutting the box and using one of the Eaton CH retrofit kits to replace the innards.  Since your box is a mere, oh, 7" wide by 11" tall, this requires an Eaton RWCH6L125N retrofit interior, the matching cover (a CRWCH6ML of the appropriate size), and the appropriate Eaton CH breakers (it looks like you can use a CH220 and a CH250 along with a pair of CHT1515s, but I'd need to know more about what this subpanel feeds in order to specify for certain).

2) You will need to run a ground wire from the subpanel to the main panel, and separate the ground and neutral bars in the subpanel; however, you do not need a GEC + grounding electrode for a subpanel in the same building as the main panel, as such an installation falls under NEC 215.6:


  215.6 Feeder Equipment Grounding Conductor. Where a feeder supplies branch
  circuits in which equipment grounding conductors are required, the feeder 
  shall include or provide an equipment grounding conductor in accordance with 
  the provisions of 250.134, to which the equipment grounding conductors of the 
  branch circuits shall be connected. Where the feeder supplies a separate 
  building or structure, the requirements of 250.32(B) shall apply.