Are the Revival Doctors still The Doctor?

Hmm... Yes, but with a caveat. I think that more recent interpretations of the character have forgotten three very important facets of his core personality, ones that exist across all of his incarnations irrespective of who they may be at the time. Throughout all his regenerations he has always remained the consummate adventurer, the occasional scientist and an individual driven to champion fairness wherever he goes. I think it's been largely forgotten that his duels with moral monstrosities were often a side effect of his travels and not the ultimate cause of why he travelled.

It's much like saying that Indiana Jones's sole purpose in life is to fight Nazis, there has been a drastic misunderstanding of the character somewhere along the line. It's part of who they are certainly, but it's not all of what they are and it's certainly not the only driving force in their life. It strikes me as being part of that loss in exploring who these people are, the Doctor's lost a little bit of his dimensionality along the way. "They're playing our song," the Eleventh Doctor says to a smiling Clara as someone is screaming in Shroud of Sorrow. Isn't that unimaginably creepy?

Nope, he's just the Valeyard, and doesn't realize it yet, but he will.

Yeah, I think so. I have some issues with how the character was written in several instances - Last of the Time Lords, The Doctor's Daughter, Hell Bent, etc. - but I also have a few issues with how the Doctor was written in the classic seasons, so I don't think that really delegitimizes Eccleston, Tennant, Smith or Capaldi's performances.

Well, let's not act like the Classic Doctors were always supremely moral: One could at times be selfish, manipulative and even willing to abandon his companions for petty reasons. Three was willing to abandon Earth on several occassions to go back travelling, Five did try to leave as soon as things looked bad a number of times, and Six could be cowardly and even bullying towards those around him. Even Seven was willing to sod morality many times and commit genocide because of a 'greater good'.

I'd even argue the New Series overemphasizes the adventurer aspect and the moralist over the scientist multiple times, especially with 10, 11 and 12.

Oh, there's definitely a sliding scale there. They all tend to jostle with one another for dominance depending on the incarnation in question. You only have to look at the First Doctor's actions in his very second story where he deliberately put everyone in danger so he could go and see the Dalek city, the Second Doctor's scheming at genocide for The Evil of the Daleks, the Third Doctor's frustration at being stuck on Earth being taken out on people like poor Jo Grant, etc. To be honest, I don't think he's really had an overtly scientific slant to him since... Oh, Jon Pertwee's Doctor, maybe Peter Davison's. It's a shame really because there's so much that can be dropped into the programme now like quantum, three-dimensional printing and the like. I learnt about the theory behind matter and anti-matter from The Three Doctors as a child.

Honestly, yeah, that's how I viewed it. There were certain aspects that got shoved right to the front because of his traumatic experiences during the Time War, The Parting of the Ways showed that Nine wasn't strong enough to make that decision again. To have the annihilation of two whole worlds on his conscience scarcely bears thinking about, I think it would have probably destroyed him if he had gone through with it. The Doctor we know would have disappeared completely.

Absolutely. If I met 20 year old me I'd want to slap him around the head for being such a tool. Everyone changes to some extent, and that's normal. So (chronologically) Hurt, Eccleston, Tennant, Smith and Capaldi are all very definitely the Doctor. You could have asked the same question when Hartnell became Troughton, or Troughton became Pertwee, Pertwee became Baker 1, and so on, especially when Davidson became Baker 2. So, the revival guys are very definitely the Doctor.

No. They're all Nick Briggs in disguise.