All we know is that Anderson was told Moriarty was dead, and that John was told that he blew his brains out.  Mycroft could be shocked that Moriarty is loose from wherever he was confined.  We saw someone who appeared to be a dispirited, mentally deranged man kill himself on the roof.

Keep in mind that in the canon, Moriarty had a brother, also named James.  If they were twins, it would be very convenient.  And of course, the one who died on the roof might have been subordinant to his smarter brother.......

Remember, there is one brief moment when you see him live, not a gif.  I'd like for him to be alive, these are Dr. Who's writers after all, and they have played fast and loose with the canon, to say the least.

Of course M is alive, and he is the 3rd sibling mentioned in passing in Vow.  He is Sherlock's fraternal twin, given over to an institution within 2 years of birth due to psychotic behavior.  Mycroft judged all children as idiots from his childhood experience with M and the "stupid" Sherlock, who wanted to grow up to be a pirate.  M came out of nowhere because he liberated himself from the asylum as soon as he was capable.

Keep in mind Anderson IS a forensic scientist.  He's not a fast as Holmes, not a brilliant, not as observant, but he uses the same methods Holmes does,  It will just take him longer to work out the solution, and he'll probably have it long before Watson figures it out.

If you take a look at the deliberate mispellings in ACD's text of the story, Holmes is obviously giving false info to Watson about what he did during the 3 years that he was dead.  As spelt, Holmes visited llamas (the andean camelids) in Tibet and did coal tar research not in Montpelier France but in Montpellier Vermont.  ACD never did come clean. (He had killed off his blasted series character and couldn't come up with believable explanation for his re-appearance, so he just faked it).

Certainly, this is very non-connonical, as is (but to a much lesser extent) Sherlock on the BBC1, and Sherlock Holmes in the 4 (so far) novels by Barry Grant set in this decade.  The challenge, and I think the fun, is figuring what an analytical genius like Doyle's would do in today's world of forensic science as a college major, near instant communication world wide,  and the many new kinds of crime we see in the 21st century :-).

Why be gone for 2 years (besides the movie contract in New Zealand and the 3 year period in the cannon)?  Doyle never convincingly answered that question in "The Empty House" (llamas are in the Andes, not Tibet, and Mountpelier is in Vermont, not France...did Doyle make these typos to play with the readers???).  The answer had better be good.

You have to put both series, Sherlock and Elementary, in context, that is, where is Holmes in terms of the Cannon's portrayal of his life.  In the cannon, Holmes starts his career in about 1887 as a consulting detective and chemist, and becomes well known later through Watson's accounts.  Reichenbach and his faked death occur 13 years later, at the height of his powers and public reputation.  In Sherlock we see Holmes just starting out, with the confrontation with Moriarty occuring within the first couple of years of the young investigator's career.  In Elementary, the storyline is definitely post Reichenbach, the fall has been not from a height, but into addiction and insanity, and an older Sherlock is haunted by the knowledge that Moriarty made a complete fool of him, and guilt from having become an addict, now a recovering addict. His old life was destroyed by the fall.   His reputation had been outstanding in London in the 13 years prior to his collapse, but he is seeing if it is even possible to rebuild his old self in the CBS series.  You have a much more tentative and fragile Holmes here.