The fastest mission runners - that is, the ships that kill the rats the fastest, as opposed to blitzing for objectives - are as follows:
1. EM/Thermal rats: Nightmare. The Nightmare edges out the Paladin unless you plan to salvage while you go (don't salvage while you go, get a Noctis if you must loot). Use Tachyons, 4 hardeners, cheap XL complex booster and shield boost amplifier. Lastly, pack on a cap injector, two tracking enhancers, and three heat sinks. The Nightmare's tracking makes Tachyon beams usable against even frigates at range, and you only need to use MF for most missions. DPS averages in the 1000-1200 range. You can make it even more deadly by dropping the shield boost amp for a tracking computer and replacing a tracking enhancer for another heat sink, but I find that I prefer the safety margin. This build requires perfect cap skills for more difficult missions.
2. Guristas/Serpentis: Caldari Navy Raven. Typical 4 hardener, setup for tank, adjust to taste. Fit a target painter and use t2 cruise launchers, switch to tech 2 hi damage cruise missile for BS targets, and regular cruise for everything else. This fit averages around 800-900 DPS, but all damage can be conformed to the target's weakest resists, making the effective DPS very high. This is the best all-around mission ship, and missions faster than a Tengu or Golem. The disadvantage of the CNR is that any build with a target painter will have a weaker tank.
3. Angels/Mercs: Vargur. I love the Vargur, so much so that I'm willing to put up with a bit of inefficiency to use them against Guristas. Use autocannons - artillery damage is inherently wasteful, and the long recycle makes popping frigs at range a no-go. Shield tank is the way to go, with tracking enhancers or tracking computers to give you the falloff you need. Pack short range and long range ammunition, and you can function effectively against any enemy. Still, this beast is most deadly against Angels.
One thing many people do not think about when picking a missioning ship is the extreme ranges some missions present you with. In terms of speed, a Golem can wipe out enemies very quickly, but the flight into torpedo range means that dps is not actually applied for quite some time. While MWD's do work in mission spaces now, they still take time to get you into range, and dramatically reduce your active tanking capabilities as long as they are drawing capacitor. So while your Golem, Federation Megathron, or Tengu (Tengu has max 78km locking range) is still flying into position, a Nightmare or Vargur may well have been blasting away for 2-3 minutes. Since a Vargur can easily complete a mission in 15 minutes, and a Nightmare in less than that, those extra few minutes add up.
In regards to drone boats for missions, I can't say that I like them. The drones occasionally get aggro'd, but the real flaw is that the DPS output is simply sub-par. A Dominix or Ishtar may take two or three times as long to complete a mission than a Nightmare or CNR.