Tough question for a tough movie. I admit that the movie also left me quite puzzled. What helped me to understand it better to a large degree was Chris Stuckmann's analysis, which I think is pretty consistent.
Now I'm not going to paraphrase the whole thing here, but the main point of it is (and there many people analysing the movie seem to agree) that both Jake Gyllenhaal characters are facets of the exact same person and the movie represents the struggle of said person with the pressure of his marriage and his infidelity and the events of the movie are not to be taken as happening exactly that way in reality. Or as director Denis Villeneuve describes it:
It's a man who decides to leave his mistress to go back to his pregnant wife and we see the story from his subconcious point of view.
But let's look at your specific points. I'll try to explain them based on Stuckmann's version and adding some further thoughts here and there based on that version (I'll also continue the practice of calling them Teacher/Actor Jake respectively, instead of Adam/Anthony):
The poster, which indicates that at least some parts of the film take place inside the head of actor Jake.
Well, it's quite clear the movie is not to be read entirely literally, as you also pointed out yourself. But since Teacher Jake and Actor Jake are probably insantances of the same person anyway, it doesn't matter so much in whose head the story takes place. I'm not sure we're really supposed to make much out of the poster at all, it's a poster and doesn't need to convey any too story/analysis-relevant information at all (let alone the question if the director/screenwriter had any input on its design at all). So all we know from the poster is, Jake Gyllenhaal is in the movie, he wears a leather jacket at one point and it's maybe set in a city and might feature things as weird as strangely deformed spider thingies. That's really all we can take out of it.
The wife's strong emotional reaction after seeing teacher Jake which speaks against a literal reading of the film.
Actor Jake's wife soon appears at History Teacher Jake's school and her look upon seeing him is one of extreme sadness and pain. Then History Teacher Jake asks her a question, which reveals even more. "How many months are you?" - "Six." - "That's nice." I find it very interesting that Actor Jake hasn't been to his talent agency for six months and his wife is also six months pregnant. This adds to the idea of Actor Jake having extreme commitment issues, most likely brought on by fear of being controlled, or dictated, by a woman and child. When History Teacher Jake gets up, she attempts to call him and the filmmaker makes it very obvious that Actor Jake answers after he disappears from frame, leaving the possibility that they are the same person out there. This is clearly not a fantasy or delusion going on inside the mind of his wife, but it is a psychological disorder of sorts for the two Jakes.
Later when she's home [...] she lays on the couch said and afraid, saying how the man she saw at the school looked exactly like him. "What's happening", she says. And when he says that he doesn't know what she's talking about he says: "I think you know." I feel that she is slowly discovering that her husband's mind has split. She wants him to come to terms with it and accept this fact, which is why she appears so emotionally bothered by it. I mean logically if you found out that your spouse had someone out there who looked identical, would you be so emotional about it, crying and everything? It'd be weird for sure but it's clear to me what she's concerned about is that they are literally the same person and she's hurt and scared by the fact that she's driven him to this state...
...This would leave me to believe that earlier when his wife says she went to his work that she was in fact starting to realize the magnitude of her husband's phsychosis and was desperately trying to connect with him, to try and help by going along with his illness, acting like she didn't know that he is a teacher in real life.
The car accident, the scar, and teacher Jake turning off the radio broadcast about the accident
The whole scene with Actor Jake dating Mary and the following accident seems to play out entirely inside Teacher Jake's subconcious while he is at home with Actor Jake's (and his) wife. It is a representation of his inner struggle to end his relationship to Mary, as well as his whole unfaithful Actor Jake personality, and stay with his wife (which successfully ends in favor of his wife, as the accident represents).
Him turning off the radio the following morning might represent him trying to forget anything that reminds him of Mary and his past unfaithful self. But on the other hand it might as well represent him already forgetting/ignoring that he ended the relationship, because the next thing he does in this scene is find a new key to the sex club he was earlier and lying to his wife in order to visit it. He is thus again falling back into his unfaithful personality.
I don't have much of an idea about the scar, it may just be a device to show that both Jake's are indeed identically equal or it might be a hint that this was not the first "accident" and that Jake went through such a process of abandoning his mistress before, as also evident from his falling back into unfaithfulness at the end.
The woman in the torn photo (who seems to be neither Helen nor Mary)
I'm not entirely sure what to make out of the photo either, but Stuckmann seems to think it is actually showing Helen (his wife) and I might agree that it is hard enough to recognize for this to seem sufficiently plausible.
He compares a noticibly torn photo of himself to the actor, probably a post breakup photo, with the woman he took the photo with torn out. [...] This is when Teacher Jake notices the photograph. It is identical to the torn one he had in his possession earlier, except this one isn't torn and shows his wife next to him.
It could thus be part of Jake's process to reconcile with his marriage, that already started when he visits Teacher Jake's apartment while Teacher Jake is together with his wife, as described above. When previously looking at the picture he was "torn" from his wife but is now starting to find back to her.
Though, there is a comment thread under the YouTube video that also seems to present some interesting approaches at explaining the photo (they might even warrant their own answers, but I'm currently not entirely sure about them).
Mary's reaction to the ring mark
This builds upon what I said earlier about the accident playing out inside Jake's head and representing the end of his relationship with Mary. Thus, Mary noticing the wedding ring mark and being shocked by it could represent the crumbling of her relationship to Jake and her realization that he is about to return to his wife, whereas previously she never saw a wedding ring (as she's actually Teacher Jake's girlfriend) because he completely ignored his wife whenever being with her.
The mother's voice message disapproving the apartment and her in-person words to teacher Jake that say she likes the apartment and that Jake should quit fantasizing about being a third-rate actor.
This just reinforces that both Jakes are actually the same person. She approves his ordinary teacher personality along with his wife and their apartment (thus actually Actor Jake's apartment in the movie), but disapproves his unfaithful wannabe-actor personality along with the apartment where he uses to bang Mary. Her earlier disapproval might thus not necessarily have been literally about the apartment, but about his other unfaithful side in general (if that phone call actually existed in reality at all), as his mother seems to be aware and disapproving of his infidelity ("you have enough trouble sticking with one woman, don't you?"). Thus, Teacher Jake's apartment might not actually exist (or it's really an empty apartment rented for time with Mary).
The spiders which also speak against a literal reading of the film.
The spiders have also been adressed before in this related question and part of the answers there support the fact that
Spiders in the film represent women, at least in the way that Jake views them. He has a deep fear of commitment. His wife is pregnant, he's been unfaithful in the past. [...] Spiders in the real world catch prey in their webs, they entangle what they are about to devour. [...] The film is in a way saying that Jake feels that he's lost his freedom, he feels trapped within the web of his marriage and the weighty responsibility of having a child as well as remaining faithful. The constant reference to dicatorship and being unwillingly ruled has to do with Jake feeling that way about his marriage and eventual family.
In the opeing of the film the narcissistic unfaithful Actor Jake gets into the private sex club with his key and watches a woman about to squash a spider. Through his unfaithfulness he longs to squash his wife so to speak, or their love at least [...] His dream of the nude woman with a spider head supports that women are the spiders as well. And directly after that a woman who was very sexy, who was dressed almost like a stripper passed him and he felt the sudden urge to follow and look at that woman [...] So how about that giant spider roaming Toronto? This scene took place directly after he met with his mother. I belive the spider represents his mother.
So how about that final spider? Why did his wife turn into a huge friggin spider? Why did the spider react in fear? And why was his look one of almost acceptive disappointment as if he has just come to terms with what his life is like? When Jake opened the package taking out the new key, he realizes it's the key to the sex club. He gets a look of temptation on his face and immediately decides he has to go there, telling his wife he has to go out tonight. Suddenly his wife is no longer answering him as he calls to her and when she is revealed to be a spider she backs up in fear of getting squashed. Jake was able to successfully eliminate the unfaithful narcissistic portion of himself, but at the first sign of temptation he goes back to his adulterous ways. This is why she reacts in fear.