:D My 2 cents:
a) How much does the galaxy know about Jedi General Anakin Skywalker, (who in old cannon was wide known as "the hero with no fear")? Is his reputation wide known?
Nothing, I think, except for veterans (like some of the rebels). When a new order is established, it passes its own "damnation memoriae", erasing every reference of the traitors (i.e.: the Jedi) of the new regime.
b) Why is Vader so secretive about his identity considering that, had Mustafar never happened, his identity would have been public? Which leads us to
c) What is all that stuff about Vader's double personality, and referring to himself in third person? Vader IS Anakin Skywalker, and he knows it in the films, he says "I am your father". When he denies being Anakin in ROTJ, it could well be a metaphor of his election of the Dark Side to Luke.
He's not secretive, he probably just disown his own republican past and assumed his imperial fighting name.
I agree that it's a metaphor: throughout the movie, there's a distinction between Vader (the dark side) and Anakin (the good/light side) of "Mr.Skywalker".
d) If Anakin Skywalker, widely known as a war figure across the galaxy officially "died" in Coruscant, how comes he is no martyr to the rebellion, or never gets even mentioned, except as a "great pilot"?
Maybe he is considered as a martyr, who knows? We've seen just 6 hours of lore :D
Or, he "died" as a jedi knight in the clone wars, unknown by most people. Maybe he wasn't even considered as a Jedi, maybe the era of the Jedi was already ending.
e) If the Rebels held Anakin's figure as a martyr, how would that impact on Vader?
I guess it would put him to shame to be known as a hero to his enemy.
f)If Vader's identity was known, how would that impact on Luke between the rebels
They would reject his legacy, I guess. The imperial propaganda would use his name to justify their oppression.
g) How comes no one outside the rebellion recognizes/associates Luke Skywalker with the other "mythical" Skywalker that freed planets in the clone wars.
Again, either Anakin wasn't known outside of some people, or Skywalker is a relatively common surname.
h) If this happened, how would that impact on Vader?
We kind of already know this: it's his son, after all, "something" he may use to take over the emperor.
i) Would anyone who knew Darth Vader's identity inside the Rebelion (read Ahsoka or ObiWan if at some point he joins for some mission in the new cannon) keep it a secret, if Anakin's figure was to be held in high regard for morale of the troops?
Definitely. I know Anakin and Vader were two different men in ANH, but that's basically what Obi-wan did with Luke.