HID bulbs are also technically xenon bulbs and there are several types of them. They are what are termed xenon short-arc lamps and the arc is run at high frequency. Flash bulbs are more like xenon long-arc lamps and are generally fired with single shots or at low frequency. The xenon is in there for exactly the same reason it’s in a xenon flash bulb as they essentially work using the same principle of electrical gas discharge. Think of HID as little flash bulbs flashing several thousand times a second. They can and do produce EMI/RFI. They are (were) fairly commonly used in desk lamps and accent lighting and used to replace low efficiency halogen units for those that could afford them. Although longer lasting, whiter light and far more efficient than halogen, they are/were far more expensive. Recent developments in high power LEDs make them the choice these days to replace both expensive HID and inefficient halogen lamps in such applications. The most common and sensible uses for xenon short-arc HID lighting now is for theater projection lamps and automotive driving lights. The automotive type typically employ metal-halide for the main light source for greater efficiency and more tunable light temperature. In these units, the HF xenon arc is usually only employed for the first minute or so until the metal-halide salts are up to operating temperature.
As suggested, you should try to move the lamp or Arduino to a different outlet and further away from the lamp.
Might be, but the HID UHP lamps used on projectors, headlights etc. are all run on low voltage DC, not high frequency.
The Xenon short-arc lamps, like used in stage lighting and cinema movie projectors run at a low DC voltage as well. 20-30V DC at hundreds of amps. They are ignited with 30Kv, and then it’s pretty much a dead short, so the voltage across the lamp is very low.