Instead of hammering the FSD, try to kill the power plant. That will usually go down faster than the FSD does, and when that goes, the ship is done.
The only way you might get the same mission as other people is by being in a group when it becomes available at the starport, and you're there at the same time. I've never actually been in a group with another player, so I'm not certain that each person doesn't end up with their own set of random missions when the BB loads. You might check out Mobius if you're looking for a PVE group, I get the impression that they're a fairly decent group to run with.
Killing those clippers in a Cobra can be done. It takes some work, but you can do it. Make sure you have countermeasures of some sort. Chaff helps if they're using turrets or gimbals. Make sure you're paying attention to power management. If you're putting all your power into weapons, shields will suffer. If you're all in on shields, your weapons suffer. If you're running all Multicannons, it's a little easier, since they don't drain the power or overheat as quickly. I used to run 4 gimballed MC and could kill the Clippers as long as they didn't manage to run and jump. Anacondas were a bit more of a challenge.