1) No. Not that I'm aware of.
2) You can. But they did not provide that option internally. But: The metronome can be heard on the monitor bus - so, technically, you should be able to connect a cable from one of the monitor outs and run that back to an input and record it. *However* - if I understand correctly - you shouldn't have to do that given that the metronome setting will be saved for/to whatever you might have already recorded with it. The only reason I can think of to record it is if you need a count-in prior to starting playing. And even then... I would probably just use some sort of percussive sound myself in time with the metronome for a count-in.
3) That's what I would do if necessary. Record a handheld metronome with a mic. Or somehow acquire some sort of drum machine or time keeping device that you can plug into the inputs to record it.
Used (and new) drum machines are pretty affordable these day. Heck... EZ Drummer 2 is not a huge purchase and is outstanding drum software that's easy to use. It has stand alone feature that you can just drop wav files onto the DP via the Audio Depot. That's how I use it!