There is a raging debate on this subject in this forum. Have you looked around a bit before posting? Your questions may have already been answered.

For my 2 yen, the answer is you absolutely should give yourself at least a night plus full day in Kyoto. If it were me, I'd do two nights in each city, with the run down to Kyoto late in the second day so you would get a full day plus a partial day in each.

If it helps, you can easily go direct from Kyoto to NRT (via Shinagawa Station in Tokyo) on your day of departure unless your flight is before about noon. Or, you could fly from ITM to NRT to connect to your international flight. I'd do the train but then I really like Japanese trains.

Can't help with your other questions as I never use high-end hotels or do karaoke.