One thing I never truly understood, even upon rewatch: in "The Woman", how did Sherlock come the conclusion to that Irene was working for Moriarty just going by her missing birth marks?
Getting a birth mark removed isn't something a captive being subjected to psychological torture would do or be able to do and, given that she had those birth marks the day before her death, the only time Irene could have had that birth mark removed was when she was a captive being subjected to psychological torture. Moriarty wouldn't care enough about a captive to either have it removed at all or to have it removed in a non-messy hospital-grade fashion (if it was truly threatening her life and Moriarty wanted to keep the bargaining chip of Irene). Therefore, Irene wasn't the imprisoned victim she presented herself as and, if she wasn't imprisoned, the only reason to allow Sherlock to believe she was still deceased would be if she was doing so at Moriarty's behest.
I don't understand how Sherlock figured out in "Dirty Laundry" that the couple were spies. The Russian traditions (though I've heard that many of them are more like folk tales) seem to be made too much of a deal of, and while they may show nationality, they may as well simply like the traditions or just happen to have some spare change around the place. I don't see how, in a post-Cold War setting, that should make them spies.
He figured out they were spies based on the videos the wife was recording from the delegates and various important figures. He figured they were Russian or at least of Slavic origin by those traditions.
Irene Adler is believed to be killed by M, which means that a huge pool of her blood was found in her apartment. If she is alive, then how did she manage to fake 5,6 liters of her own blood, and fool Sherlock with it?
She extracted (or had someone extract) her own blood, spaced out to give herself time to replenish it for future extractions, gathered enough that finding such a volume of blood would make it nigh impossible for any police authority (even Sherlock) to consider her alive and kept it in cold storage until she had Moran bring it over and dump it on her floor. Boom, instant crime scene. It's a super common way of faking your own death in fiction.
They had to make some tests, and Sherlock should be smart enough to notice that the blood wasn't fresh, but stored for a long time in some cold place.
If you consider that Irene was the most important person in his life and she just disappeared when the blood was all over her living room, his emotions dominated him and he was unable to make a precise deduction. Also, she always had contacts within the police. Sherlock is not the one who does the lab tests. She could very easily fake a result using her human resources.
Remember the locked room and her secret project? It's possible she was storing everything there, blood included.
Why is Elementary!Mycroft so utterly different from his canon self? There's no evidence of him being able to outdo Sherlock in a Sherlock Scan competition. He isn't an unacknowledged cornerstone of the government. He isn't antisocial. He isn't so utterly fixed in his ways that seeing him anywhere other than his residence, his job, or his club is a spectacularly unusual occurrence. The only thing Elementary!Mycroft has in common with the original is the name and his (former) weight.
He's set for a recurring role in the second season. It's likely the other part of his life will make an appearance then. Perhaps his French conversation was government related.
And even if the above statement doesn't come to pass, although it's quite possible, this is a show that has not been the most faithful to canon versions of the characters, Joan and Moriarty for instance.
Also, canon does mention that Mycroft is lazy (or at least Sherlock perceived him as such).
It's also possible that, like Elementary!Lestrade (Who is shown to have been exactly like his canonical self until Holmes left London and he ended up as the Deconstruction of his own character he is now), he used to be like his canonical self until Holmes left London His severe cancer being to blame for such changes.
In the books, Mycroft did some light work for the government. The accounts. Occasionally. It was extrapolated in adaptations that he was a precursor to M/the British Government/Mi6, because that gives his character a bit more to do. It appears Mycroft has been Lost in Imitation.
"You are right in thinking that he is under the British government. You would also be right in a sense if you said that occasionally he is the British government." Sherlock on Mycroft, The Adventure of the Bruce-Partington Plans. Though he wasn't in intelligence. His job was to be omniscient. If an issue concerned multiple departments, any number of people in each department might be able to explain how the issue impacted them, but Mycroft would be able to explain that for all of relevant departments, and how their interests and issues interacted to form the big picture.
It actually looks like they transferred several of Mycroft's original characteristics to Holmes' father (his emails even list him as M. Holmes).