Does repeated turning on/off Xenon HID headlights harm them?

Yes, HID bulbs do not like excessive cycling, it causes damage to internal contacts as the arc is struck with high voltage. After striking and once a stable arc is formed, the voltage drops to around 40-80V. The bulb lifespan is still very long, even if you switch them multiple times per day.

HID bulbs do wear and burn out. Acceptable life span is between 5-10 years, depending on how much the lights are used. The first sign of failure is reduced light output, second is colour shift (colour of the light starts to shift to red/purple), third is flickering/random on-off cycling. If you see any of these symptoms then the bulb is due for replacement.

I have replaced HID bulbs twice in my life: once on a 13-year old Saab and once on a 2015 VW Golf. The Golf switches it's headlights every time you unlock the car, which contributes to bulb wear. The Saab had (mandatory in Europe) DRL function, where the lights would always turn on when you started the car and stayed on until you shut down the engine.

I'm no expert, but I have the same setup in my car. I had to install a capacitor to give the HIDs constant power so they wouldn't get power at 60Hz. However, they do turn on and off once in a while, say going through a tunnel.

I haven't had a single issue with them in the last 2 years - since they were installed. As long as they have proper power when they're needed, I don't think it's a big deal.