I hope they do it. I think hair colour and we'll, anything to do with appearance might be harder. Too many variable. What if you picked a more brown sort of red, but it's red to you and it doesn't register?
Race. We should be playing as human only so that shouldn't be a problem.
Class. Absolutely it should happen. I believe it happened a little bit during the trilogy. Dr Chakwas mentions your implants or something, but class should have more of a role in your role play. Not just to define your combat style.
For hair color honestly that could have been done easy in DA:I, I mean we didn't had 'TS3' CC in where we could go wild with hair colors, so dialogue or two could have existed from Iron Bull in where he is like "Hey I just realized that you were redhead". So this little smart programming wasn't hard to do, I just pointed it out because it was missed opportunity in here.
For second part, race or no I was hinting on our possible character backgrounds(in case we have more of them). For example in ME if you were colonist who's family and friends got killed you could have saved that girl in ME1 from suicide, if you were from military family(spacer) you would meet that one guy in ME1 that worked with your mother and also there was message from her in ME3(don't recall was there something special for 'earthborn' background. So that was my point if we get once more multiple backgrounds I would like they actually have some impact now and there, otherwise in that case I can live with Bruce Wayne personality like background.
In ME3 they were doing good in 'Omega DLC'...at least as far as engineer is concerned, you finally get to feel a little as your supposed class, also when Liara comes to make hologram of you she also acknowledge your class. I found that nice and honestly it would be even way better if this was more implanted into upcoming game. I think at least we can all agree we wish to be class we chose not for example adept by choice and soldier for real by game.
Side Note: Domino is awesome 
I don't see these two as a problem. Just because a character asks questions about something doesn't mean that the character doesn't already know the answer.
It's a good thing that Hawke can be hypocritical. Imagine if he couldn't, and was required to state all of his opinions earnestly.
Actually these two were biggest problems!
For one reason, all races in DA:I felt like "I'm human regardless of what race I chose to play" and in DA(except in DA:I) all of your chosen specializations also felt like "We don't exist" because they were never recognized. DA:I is pretty much first one in where people are like "Oh so this is what you are", until then you could have used blood magic in front of all templars and no one would bat an eye in here.