thanks for joining the conversation, and the series of pictures that you took time to add to your post.

I will give it my best shot--
Q#1--By all means timbers were hewn by hand to a square dimension, that is close to what was needed, for most purposes that is the way they were used, but in some cases for example exposed timbers in a home, or in many old country dutch barns the special beams like the anchors and their posts were adze finished. the surfaces will show the unmistaken marks of the adze blade, and should if done properly be undulating in appearance, and no hewing marks will remain.

In exposed ceiling timbers, say in a early log, or timberframe home, after the adze has did its job, the lower edges would be beaded by hand to round out the appearance, when good workmanship was the name of the game.

In some swing beam barns the swing beams only were adze finished because I suppose they were the main focus of attention.

As far as your photo I would say that the timber was a bit off square, and with the timber laying on the ground and standing on top of it, an adze was employed to remove the excess wood on the lower corner. You will notice that not much was removed on the upper corner, and the tell tale rough scoring marks (2) remain slightly visible, and some of the regular scoring marks. The tradesman who did the adzing used an overlapping type of repititious swings and created an unusual appearance, in my opinion it is definitely an adze that did the final work, what do others think?

Q#2--In my opinion axes with less than 9 or 10" in width would not qualify to be called hewing axes but rather hatchets, what you may have seen was timbers squared with a regular felling axe.
All the hewing axes in this area would have been 10 to 14" wide on the cutting edge and have only one bevel, the side towards the timber would have been flat, now just a minute--nearly flat, and I think that this may answer another of your questions, the flat side of a hewing axe is not flat but slightly rounded and will be only noticeable if placed on a flat surface, this is for a reason the centre if the blade will take wood nicely, and not bite in at the edge. A broadaxe in an experienced hand will produce a surface very near an adzed surface, and will have an undulating surface texture, and be hard to tell the difference. I have pondered over many timber finishes to try and acertain what historic tool in fact created the final finish, not an easy call sometimes. this type of scrutiny only comes into play when doing an exact restoration and having to put on new timbers an original finish.
Q#3--the company that produced so many axes no doubt was shipping internationally, and knew what type of axe the population was using in that country, so it required a great variety of styles to be welcomed as an import.

#4--You are right on the money I would say the joint was formed by all means using and adze, if I was doing that joint it would have appeared very similar, and I would have used an adze rather than chisels and slicks

#5--I am not familiar with gunstock joints so I will leave this up to others to comment on but nice pictures

They say a picture is worth a thousand words and if there are other examples out there that need to be commented on please lets see them.

Please don't be afraid to comment on my best shot at an explanation it is by friendly chatter that we can learn, and I am always willing to learn, and I want others to also.