As far as understand it the colour temp of the ambient lighting will not be relevant in regard to the brightness setting as the calibrator is colour temp blind in respect the impact of the ambient light on the calibration process.
I know. Never said it was.
However it may likely affect your perception of the colours you see on the screen during the pp sessions and as such more so when you get your prints back to examine them!
Not may be likely.. but almost certainly will.
In regard to viewing/examining prints. I have a hobby light that uses a lamp that is 6500k I.e. the same colour temp as the screen but I do recall reading that others would also check that they look 'right' lit by lamps of 5500k as that light is more common in galleries and domestic situations. On this latter I would welcome further feedback
Most print booths are D50, yes. That's not because it's closer to the lighting in galleries. I've been in thousands of galleries, and I've never seen any with D50 or even D65 lighting. D50 lighting historically was chosen because it more closely matched the colour of paper bases, but it's no longer regarded as essential... just one of those legacy things that persist. In fact, most professional print booths can be
bought in D50 or D65. Ideally, D50 should be used for proofing prints, but there's no technical reason that makes a great deal of sense, especially when the same international and European standards also apply to D65 and so many paper bases use optical brightners and are sometimes a very cool white. It's far more important to be consistent throughout the whole workflow.
The colour of your room lighting hugely affects your colour acuity. For example... I don't have black out blinds in this room here, as I rarely have time to edit during the day. As a result, when I am in here during the day doing other stuff, the apparent "colour" of my monitor shifts greatly depending on the outside colour temp. At sunset when the sky is very warm, the screen looks blue/green despite being perfectly calibrated. On extremely cloudy days at twilight when the colour temp soars, then it starts to look very warm/yellow. If I tried to edit a picture under these conditions, I'd almost certainly be dialling such colour casts out. At sunset for example, as the screen looks blue/green, my images that I try to visually balance would end up being red/magenta in bias. So, if you calibrate to D65, then have normal 3200K domestic lighting, your screen will appear to be very cold. As a result, you'll over compensate and produce very warm prints.
You need the room lighting and calibrated white point to match. That's NOT to say however, that you should calibrate the screen to 3200K and just use normal lighting. That's just too low, and will never be perceived as neutral... just as your domestic lighting is never perceived as daylight. You're always aware that it's warm/yellow. There's only a narrow section of visible kelvin temp scale we can "normalise" ... between 5000 and 8000K. Anything lower than that will always be perceived as warm/yellow no matter how long we sit in that room at night, and conversely, anything above that limit will always seem cold. Within those limits it's possible for the brain to normalise the lighting so it just appears neutral. The closer to the centre of those limits you are, the easier it is for the brain to do this, as that is closer to the temp of actual daylight, which hovers around 5500 to 6500k most days at mid day. Actual daylight (meaning the light being emitted from the sun is 5780K, but our world is not lit ONLY by the sun, and when factoring in a blue sky, the added diffuse light being scattered raises this temp to between 6 and 6500K depending upon the sun's position. This is why there's always some debate as to what constitutes daylight.... 5500K or 6500K. Daylight is highly variable though... daylight from a north facing window (Northlight) is closer to 10,000K on a clear day.
Calibrating your screen is only one aspect of good practice. Controlling the room lighting is another, and if you print.... decent media calibration and finally print proofing is the final part. So if you don't print yourself, you're still responsible for two of those things: Screen calibration and room lighting.
Monitor calibrated to D65....
a 6500K room light... and a means of blocking out daylight during the day will ensure pretty consistent results.
Those pro-lite lamps I've linked to are not only cheap, but pretty consistently good as well. Colour temp is as advertised (Yes I've measured it) and they start instantly, and don't take ages to warm up like most (except when very cold)