I have to caveat first that I am not an automotive engineer; my comments are based on my own observations in my cars. Same.
The rise in tyre pressure appears to be linearly correlated to the rise in tyre temperature. A 10 degree Celsius rise in temperature typically increase the tyre pressure by 4 PSI (0.2bar). The tyre temperature will rise regardless of the weather, due to friction with the road surface. 1) Correct. However, the pressure can be different because the 4 tyre temperature can be different.
I don't think there is a specific 'operating' temperature of the tyres, i.e. tyres still work in cold climate v.s. warm climate. I guess you are inferring 'optimum' temperature where the tyres may provide the best benefit (grip, fuel efficiency). This seems to be a range of between 2-8 PSI increase from cold tyre pressure. I derived 2-8 PSI from driving in different periods of the day. 2 PSI increase is at night, where the ambient temperature is low, and the rise in tyre temperature is mostly due to road surface friction. 8 PSI is when the road is extremely hot under a hot sunny day, where I observe my tyres temperature reaching 50 degrees Celsius. This refers to normal day-to-day driving, instead of racing tyres, as I understand racing slicks do have operating temperatures for optimum grip on the track. 2) That is why the question on cold tyre temperature. Cold tyre temperature is in relation to the ambient temperature at the time of use. In your scenario, there will be many cold tyre temperature. So effectively, when the day is hot, there will be over-inflation if the cold tyre temperature is in the early morning. This is looking at the details but if not looking, anything also can. The challenge is finding the balance.
The issue with pumping tyres while they are hot is that there is no reference point to tyre pressure vis-a-vis tyre temperature for the 'compensation'. For example, if you were to pump your tyres at a petrol station while your tyres are hot, you can only see what is the current type pressure, not the tyre temperature. You cannot provide an accurate 'compensation' because there is no information whether the tyres are hot @ 40 degrees Celsius, or hot @ 50 degrees Celsius. When the tyres' temperature drops to ambient temperature, your tyres may not be at the recommended cold tyre pressure. 3) The tyre temperature will affect the pressure. If the tyres are pumped the same pressure at cold, the pressure at hot could be different because the tyre temperature could be different. So when it reaches back to the cold temperature, it will be different. So the question now is, which is better, e.g. front tyre, one side higher pressure then the other ? The compensation is based on one cold tyre pressure and not multiple. There is no solution to multiple cold tyre pressure except to average.
I can't see how tyres can be slightly under-inflated if you pump a specific pressure (say, 32 PSI) at the cold tyre pressure. It is still 32 PSI regardless it is a cold rainy day or a hot sunny day. 4) As mentioned by you in 1), it applies to drop in temperature as well. In moderate to heavy rain conditions, the rear tyre pressure will drop if the cold tyre pressure is higher.
To me, the difference of cold tyre pressure on cooler months of the year compared to hotter months of the year is not significant. The tyre pressure will rise regardless once you start driving, and it would be within the range of optimum temperature. 5) If it is not significant to you, then there is no difference. As you have stated in 1), there is a difference.
I also understand that front and rear tyres need not be of the tyre pressure, that's why I did not bring it up. B-segment cars (Swift, Jazz) that i have driven tends to have higher pressure for the front tyres. Recommended cold tyre pressure for my Swift was 36 PSI front, 32 PSI rear, while Jazz is 35 PSI front, 33 PSI rear. D-segment cars (Mazda 6, Merc C-Class) that I have driven have the same recommended cold tyre pressure for all four tyres. 6) AFIAK, tyre pressure is dependent on load. Usually the front will be of higher pressure because of the weight of the engine. The vehicle placard will have information. I did a search for Mazda 6 placard, front is 32psi up to 4 person/38psi full load and rear 32psi up to 4 person/45psi full load.