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I've never really worked with palm trees. I know their trunk has a different structure than other trees, so I'm not sure air layering would work. I'm leaning towards "probably not". If a parlor palm is what I think it is (English is not my native language, and vernacular names are always... imprecise), I think your best hope is waiting for suckers to grow at the base.
I would to it in stages - root and cut off one part after the next. It will take three times as long, of course, but since you have to cut the branch partway, I would worry about it breaking, especially at the cut closest to the trunk, since there'd be a lot of weight on it.
No, you leave it in, to keep the cut from healing up again. (Since the cut is what "motivates" the plant to grow roots.)