I guess I'm upset that no edition of D&D or D20 hack does what I want. I think Basic Fantasy comes the closest, but its still very far off.
I want something very fast, story-focused, and intuitive that focuses on exploration and role-playing. I want combat to be quick and gritty, character creation to be minimal, and the rules system to be smooth and balanced. Characters should start out very simple and similar to each-other, but gain complexity and originality as they advance.
I think one of the main weaknesses of D&D 4e is trying to make characters complex and distinct right from the start.
Ok, so let's look at those parameters.
I want to make the game retro, but how retro should I go? I don't want the game to be a clone of AD&D, but I want to recapture the "low-powered heroic romp" feel.
I don't think there is such a thing as "too retro" if that's what you want. Are there specific aesthetics you find attractive? Do you like charts? Black-and-white line art? Hirelings? Traps?
How close should I go to the D&D concept? I the game to have a heroic fantasy theme, but should I have the exact same concept as D&D?
D&D is already D&D. Even if you make a better game, you'll never make something more D&D than D&D is already. That said, focus on the experience you want. It's pretty much a given that other people will use a game for things for which it was not intended, which is fine. But for you, the game should be focused on concepts you find appealing. Your engangement of those themes is what will make the game interesting, or not.
How should I advertise the game? I'll give away the game for free, and I'm wondering how I could advertise it in a way that can compete with D&D.
There's no one answer. If you want to crack open the market, giving away free, no-art copies electronically and charging for full color PDFs and print copies has done pretty well for a lot of people.
Another answer might be, develop a concept so compelling you can market it to an existing publisher who already has good market penetration and currently has no D&D clone of their own on the market.
Another might be, "You can't. Find your own audience."
How should I handle combat? I want combat to be less tactical than 4.0e, less tedious than 3.Xe, and still not mindless. How could I pull this off?
How rules-heavy should I go? I want the game to be less crunchy than 3.Xe and 4.0e, but I don't think I want it to be "rules-light" either. How should I handle this?
You could get copies of Warhammer Fantasy Role-Playing Second Edition, GURPS 4e, and Hero System (4th, 5th, or 6th edition, your choice), True20, and Big Eyes Small Mouth d20 (available as the Anime d20 SRD as well). Each of these games takes a different approach to flowing combat. All work really, really well without being bogged down in details.
"Less crunchy than 3e" probably means eliminating as many special cases as you can. Every rule you can identify as something you don't like, consider whether you can live without it entirely. Basic D&D has a combat section eight pages long; in Pathfinder, the equivalent section is 24 pages. Every one of those sixteen pages represents a group of cases that are addressed or not addressed in each game. "Like Basic D&D, but with more fighter options and rules for disarming and grappling" would be a good starting place. If you find yourself inventing a lot of new words, you've probably gone too far.
Also, identify what you would consider "boggind down." 3e tends to slow down because of special AoOs (the basic ones are simple and have really been in D&D since the beginning), spell buffs and situational effects, and multiple calculations that have to be done per attack. As far as being less tactical than 4e, the easiest thing to do is to write combat in English, rather than in mapspeak or specialized vocabulary.
Does any of that help?