It sounds like you have several challenges and objectives with what you're trying to do. First off, you want to video a live band in a small club. Next, you want to record the audio of that band. Third, you want to sync the audio to the video in post and edit into a video of some sort.
Recording the audio of a band in a small club is not the most difficult thing but also not the easiest if you want to do it right. I've done it hundreds of times from something as simple, yet challenging, as lavaliere microphones in a baseball cap on my head to mics on stands in the audience to sound board patches as well as multi-track recordings of all the mics split directly from the stage into my own pre-amps. So I think you have to ask yourself what kind of recording do you want. Is this simply to document the band? Then the Zoom recorder with on-board mic would probably be satisfactory. If you want to step it up a notch, then go to an external mic plugged into the Zoom. The next level would be external mic with external pre-amp and possibly an external A/D converter. My point is to get you to think about the quality of the recording and make your decisions from there.
So ask yourself if you want to record the band directly or if you want to record the sound of the band in the room. To record directly, you can either split the stage mics or use your own. Another option may be getting a sound board patch from the front of house mixer; but depending on the size of the room, this may not have all the instruments and may not be that good of a mix. If you plan on recording the sound of the band in the room, then you'll be getting more of the room sound, which could very well be boomy and really heavy in the low-end. (This is very typical in small clubs.)
But also ask yourself if you want the sound to follow your camera perspective or if you want it to remain stationary, thus resulting in a consistent stereo image. If you want it to follow your perspective, then the mics have to travel with you. If not, then you can setup stage mics or somewhere in the middle of the room and record into an external recorder. This would allow you to move around at your will, while at the same time keeping the stereo imaging consistent.
When I was recording bands a bunch, I typically used a Shure VP-88 stereo M/S microphone into an Apogee Mini-Me pre-amp and A/D converter. From there, I recorded onto a DAT recorder. The Shure mic was a great sounding stereo microphone. I also owned Neumann mics, but tended to use these very selectively. The Shure was my go-to mic for a simple stereo recording. (Kind of ironic considering the Neumann's were triple the cost.) Anyway, I would probably stay away from omni-directional mics. They're going to pick up everything in the room, including what's behind you. So some sort of stereo mic or a pair of mono mics in an X-Y pattern would be good for recording the room. I would probably stay away from shotgun mics unless you're going to use them in the traditional sense of an interview. But with that said, I have used shotgun mics for band recording, but they were in larger venues and aimed at the speakers from a couple hundred feet away.
Regarding Final Cut Pro and Premier Pro, they are very similar programs. Each has their pros and cons, but they both have about the same learning curve. What I feel is the advantage to Premier Pro is that the application ties in with other Adobe applications, which will allow you to do many other things. Also, I don't recommend using FCX. Apple seriously screwed the pooch with this application. It's more like an upgrade to iMovie instead of a new version of Final Cut Pro. But then again, it may suit your needs just fine.
As for syncing the audio, there is a plug-in called PluralEyes that can be used with either FCP or Premier, although I've never used it. http://www.singularsoftware.com/pluraleyes.html
Also, check out this link to get a bit of info on recording and syncing the audio for the 5Dii. There may be some useful info for you. http://www.sounddevices.com/notes/cameras/5dii-audio-performance/
And one last comment regarding 300m audio cables. In all the years that I recorded bands, I never needed audio cables at 300m (984 feet?). In fact, I owned a 100ft 24-channel snake, and I rarely needed its full length. With that said, if you're planning on recording the band somewhere near the middle of the room near the mixing console, your cables really only need to go from the mic to the recorder. (20 feet?)
Good luck and have fun!