Thanks for the ideas. The results of voltage measurements:
Battery with everything off: Little over 10
Battery after turning on ignition key: less than 1
Battery with starter button pressed: around 1
After starting, in N and negative cable disconnected, measuring across cables:
Idle: Little over 10
Increase rpm to around 1300 (guessing by ear): Little over 13
Increase rpm over around 1300: Drops to 6
Revving with battery disconnected does not solve headlight issue, it still reduces brightness
I went ahead and replaced the battery since it is a few years old. I also increased the idle rpm to around 1300 or so, at the point where I measured the max volts on the cables. This solved the starting and headlight problem for now.
However, It seems to me that I have an alternator problem or maybe a ground problem.
Thanks for your help!