1. If your player is willing to accept that the standard assumptions about dragonborn do not apply in your campaign, and that his character will be viewed with suspicion and occasional hostility, would you allow him to play a dragonborn?
No, he would have to play a member of one of the seven 'free peoples'. All other races are servitor races, and are presumed (or in fact are) to be mentally enslaved to a particular diety or otherwise manifestations of a particular dieties will. In any event, a character from a servitor race would take crippling xenophobia penalties on social interactions, if monsterous would be treated as a monster, and would quite concievably be in a position where I would feel inclined to dictate his behavior (in the form of 'your diety is controlling your will'). IMO, telling a player how his character should behave is the big no-no, so playing a servitor race just brings up all sorts of problems I don't want to deal with. This is one of the primary campaign elements and I'm not going to comprimise on it.
2. If your player just wants the mechanical benefits of playing a dragonborn, and is willing to accept that his character will be viewed with suspicion and occasional hostility, would you allow him to play a "half-dragon lizardfolk" that had the same racial traits as a dragonborn?
No. Granted, I don't even know what the 3e racial traits of dragonborn are, but the player was someone I trusted I might let them play say a Dwarf with draconic ancestory and use the same racial traits as dragonborn. It's worth noting that whether we altered his base racial stats or not, he could take bloodline feats as a Drawven sorcerer that would let him for example, breath fire, acquire a slightly draconic appearance, and so forth. If he took the Blood Sorcerer feat, he could even be a pretty decent 'gish'. But he would be a drawf (or whatever race) for the purposes of the game.
3. If your player wants the mechanical benefits and the "proud warrior race" background of playing a dragonborn, and is willing to reflavor his character's appearance so that he looks more human-like (perhaps he is from a noble family that has a draconic bloodline) would you allow him to play a human-looking dragonborn?
All my races are proud warrior races. Anything that isn't a proud warrior race gets extinct in a hurry. But if he wants 'proud warrior race' as a primary attribute, it just screams Orine to me. Again, I have no idea what the mechanical benefits of being dragonborn are, but I'd be willing to comprimise slightly on his base racial traits. I really would think it is unnecessary, but if I had a problem player holding up the game I'd do what I could to get the game running for the sake of everyone else.
If you would agree to any of the above, do you see it as part of your responsibility as a DM to highlight the possibility to the player? Do you see it as something that a good (or nice) DM might do, but is not actually an obligation on his part? Or do you think it is solely the player's responsibility to ensure that his character fits into the DM's campaign setting?
As a DM its my responcibility to help a player understand any of the mechanical options available to them, and to help steer them to something which inspires them and that they find enjoyable to play. It's my responcibility to try to put together some options that is as close as can be managed to any reasonable idea that the player might have. Those are my responcibilities during character creation. But I'm under no obligation to make any particular mechanical advantage or particular flavor available to the character. Ultimately, I think I can overrule just about anything. The idea situation for me is that a player new to my game acts as if he's never played before, and that he's being introduced to the game for the first time with biases, preconceptions, or baggage and instead brings to the table all the guilessness, trust, wonder, couriousity, and zeal of a new player. It's the players responsibility to create a character that is within the scope of the setting and which is entertaining for himself, the other players, and me. If he really honestly thinks that he can't create an entertaining character who isn't a dragonborn, then I'm probably going to agree with him that he's better off at some other table. I would consider that more of an admision of failure on his part than a failure on my part. I would consider it a failure on my part if nobody at the table thought that they could create an interesting character.