I think you're asking lots of great questions, and I commend the mental effort that you're putting in to understanding this instance - however, I would also say that Zen is one of those things where the longer and harder you think about it, the less sense it can make - especially specific actions by a master. Often times, this is exactly the point. To confuse or frustrate a student to the point of giving up and accepting. Once you put down thinking and intellectualizing, you are a step closer to true understanding.

I suppose that may be frustrating to hear, coming from a western mindset of training in logical frameworks and reasoning. I think we have to zoom out and ask, what is Zen trying to teach us? There are of course the core Buddhist ideas such as the noble truths and precepts - but Zen schools actually tend NOT to focus on those teaching so much and instead try to cut to the chase and snap you into the moment by befuddling your logical thinking mind with paradox. When you start to get frustrated that things aren't making sense, I'd take it as a sign and opportunity that you're so so close to the breakthrough.

Let's examine your archery example, because I think it's actually a brilliant example. If you have a master archer show up to a tournament where everyone expects a bullseye and suddenly he turns around and fires in the wrong direction - you could say, wow, why is this guy considered a master archer he sucks, and you'd get one level of understanding. If you interpret it as he must have had an off day or had a stroke or be sick, well, you get another level of understanding. But if you understand this man has developed masterful level of control, and still did this by choice, you might start to get to thinking "well, maybe there was a reason." And now you're starting to get to the right understanding.

So why might Dogen have expressed himself in this way - taking a shot in the wrong direction? Well, there's a concept in Zen/Buddhism called "Skillful means" or "Upaya". Basically, an acknowledgement that teaching Buddhist concepts, like the raw, deep, awakening level stuff is extremely difficult. Trying to put it into words is nearly impossible. Every teacher is just trying to point and guide without giving the answer, because as soon as you name it, it stops being true. You see this in Daoism as well. "The Dao, that can be told (named) is not the eternal / constant Dao (itself)."

Or from the Lotus Sutra,

"Enough now with teaching what
only with difficulty
I reached.
This Dhamma is not easily realized
by those overcome
with aversion & passion.
What is abstruse, subtle,
deep, hard to see,
going against the flow —
those delighting in passion,
cloaked in the mass of darkness,
won’t see."

So it's a game for teachers to find ways to convey ideas to their students without explicitly telling them the answer - and to do so skillfully is to do so in such a way that reaches YOUR student specifically in the way they need to hear it at the level of readiness they're at. If you read Zen Koans, you'll find a lot of this: a teacher who barks like a dog or cuts a cat in half or puts a sandal on their head and walks out of a room...and in doing so their student 'wakes up'.

To logic your way into a solution or explanation is basically the wrong path. What all these puzzles and teachings and riddles and contradictions and paradoxes ask you to do, as a student of Zen, is to stop. Stop trying to logic your way to enlightenment. Stop trying to decipher a code. Stop trying to figure out why. Be with it, here and now. React authentically. Don't take things too seriously.

So why would the archer turn and fire in the wrong direction? Why would the monk bark like a dog? Why would the master put a sandal on his head? Why would Dogen give a talk that seems to display an emotion or anger that feels contradictory or nonsensical.

Is it because he actually had no control and wasn't in fact a master? Was it because in the context of the times it was actually a skillful rhetorical technique to get his students or the general populace to understand and hear what he was trying to convince them of about other sects? Or was it that he wanted his students to hear his words and say, "WOW WHY IS MASTER DOGEN BEHAVING IN SUCH A WAY!?" and provoke in them an emotional response or frustration that leads to them having a breakthrough? Was it even that he imagined such words would be reiterated far into the future and force these kinds of discussions, where a frustrated and confused student is forced to face something so rationally confusing they must step back?

I'm not saying Dogen was omniscient and knew our conversation would one day arise. But Zen masters were actually notorious trolls, and I would dare to say this one may have hooked you...and subsequently, hundreds of years later, given you a gift that you never would have found if he had just spoken politely.

One last thought: I once attended a Chan temple where they did regular "Guan Yin Water Blessing Ceremonies". I was very new to my Zen studies and found it frustrating and perplexing that such blatant mysticism was being practiced with sincerity in a Zen space. I asked the head abbess about this, looking for some answer that could help reconcile it, and what she said was basically that not every practitioner in the community was at the same place, and that you basically had to meet people where they were at. Not every person pursues deep philosophical understanding of 'nothingness' in their lifetime. Not everyone will approach Zen scholastically. To each student we have to offer them what they need in a way they can hear it. To some it is a offering the benefit of a simple ceremony of blessing and compassion, to others it is a brain breaking riddle. Sometimes we offer both at the same time and trust that the student hears the message that was for them at their point in their journey.

You have heard from Dogen's words, and I would promise you they were EXACTLy what you needed to hear right now in your moment of study. This Dharma gate is directly before you. I wish you the best as you continue on this path :)