Harry is Dumbledore's hero and 'heir' if you will.

I interpreted this (with p<0.1) as foreshadowing Harry being Voldemort's heir, and Harry being Voldemort's idea of a better self (with power Voldemort knows not).

1) the last bit seems like Dumbledore in senitmental yet serious and regretful wise wizard mode. Harry is Dumbledore's hero and 'heir' if you will. Does Dumbledore wish he had some of Harry's cold intelligence? Does he think many of his dead friends and dead family could have been saved by someone who was harder than he was from the very beginning?

I think you are forgetting the context here of MoR: Godric Gryffindor has been set up as a quasi-emo/existentialist Hero, and heroism a painful uncertain path to travel, with countless sacrifices along the way. A life spent fighting and sacrificing beats out of one vanity and arrogance and certainty, as indeed Dumbledore himself has lost conviction and certainty. To quote the relevant passage in ch43:

Godric had defeated Dark Lords, fought to protect commoners from Noble Houses and Muggles from wizards. He'd had many fine friends and true, and lost no more than half of them in one good cause or another. He'd listened to the screams of the wounded, in the armies he'd raised to defend the innocent; young wizards of courage had rallied to his calls, and he'd buried them afterward. Until finally, when his wizardry had only just begun to fail him in his old age, he'd brought together the three other most powerful wizards of his era to raise Hogwarts from the bare ground; the one great accomplishment to Godric's name that wasn't about war, any kind of war, no matter how just. It was Salazar, and not Godric, who'd taught the first Hogwarts class in Battle Magic. Godric had taught the first Hogwarts class in Herbology, the magics of green growing life.

To his last day he'd never been able to cast the Patronus Charm.

Godric Gryffindor had been a good man, not a happy one.

Harry didn't believe in angst, he couldn't stand reading about whiny heroes, he knew a billion other people in the world would have given anything to trade places with him, and...

And on his deathbed, Godric had told Helga (for Salazar had abandoned him, and Rowena passed before) that he didn't regret any of it, and he was not warning his students not to follow in his footsteps, no one was ever to say he'd told anyone not to follow in his footsteps. If it had been the right thing for him to do, then he wouldn't tell anyone else to choose wrongly, not even the youngest student in Hogwarts. And yet for those who did follow in his footsteps, he hoped they would remember that Gryffindor had told his House that it was all right for them to be happier than him. That red and gold would be bright warm colors, from now on.

And Helga had promised him, weeping, that when she was Headmistress she would make sure of it.

Whereupon Godric had died, and left no ghost behind him; and Harry had shoved the book back to Hermione and walked away a little, so she wouldn't see him crying.

You wouldn't think that a book with an innocent title like "The Patronus Charm: Wizards Who Could and Couldn't" would be the saddest book Harry had ever read.

Harry has said that he is surprised that Gryffindor would permit his heir to be people as frivolous and humorous and chaotic as the Weasley twins with all their youthful hijinks. To which Dumbledore replies,

"Only a man exceedingly proud and vain, would believe that his heir should be like himself, rather than like who he wished that he could be."

Did Gryffindor wish his successors to be as grim and sad as him, did he wish to live such a life except as his war-torn world forced him to choose? No, of course not.

rather than like who he wished that he could be

The twins are happy and enjoy their life.

who he wished that he could be

Perhaps another time, another place, another life, Godric...

if he has the body he lied about it outright and without hesitation

But not if he only has her brain.

"Only a man exceedingly proud and vain," Dumbledore said quietly, as he turned back to the Floo roaring up again with green flames, "would believe that his heir should be like himself, rather than like who he wished that he could be."

That leaves out a third, and more sensible, alternative, which is choosing an heir which is the best person available. That person isn't likely to be an exact match to either the person who's doing the choosing or what that person thinks of as an improved version of themself.

Is Quirrel exceedingly proud and vain? Does he want Harry to be his heir?

Harry insisted on a yes/no answer to whether they're "something like a(n) Heir of Gryffindor". Since Godric did in fact want any number of people to have access to his sword, and this is something like naming heirs, Albus gives up on correcting Harry's categories for the moment.

I think it's important that this comes just a few chapters after Dumbledore regrets resenting Harry for having spent his fortune to save Hermione, when he (Dumbledore) chose not to do so to save Aberforth.

"I - I'm sorry, Harry - I -" The old wizard pressed his hands to his face, and Harry saw that Albus Dumbledore was weeping. "I should not have said, such things to you - I should not, have resented, your innocence -"

(Ch. 84)

I think it somehow relevant that Dumbledore said, "Fred and George Weasley are the Heir of Gryffindor", not "Fred and George Weasley are the Heirs of Gryffindor".

Admittedly, I don't know what it means, but at this point, I think we can safely say that even minor quirks of grammar mean something in HPMoR.