A lot of these questions I'm dealing with my fiction, Of Blood, but I'll take a stab at them.
A lot of fiction seems to portray vampires as immortal (until a steak is driven through their heart), or at least non-aging. Do you think this concept would play into the Potterverse at all?
Of course. Their immortality sets up their 'age.' So thus some of them could be technically around the time of the Founders. As well, like Nicholas Flamel, they have a vast history so that can be played up.
Does a simple bite make one a vampire, or is there more to it than that?
This is really up to interpretation. A more traditional interpretation is to be bit once and be infected with it. This is seen with the modern zombie as well where the disease rages uncontrollable; however, unlike the zombie, vampirism would not set in until nightfall. However, Rice's vampires are transformed into vampires by being bit and then drinking the blood of the vampire. And then Hamiltion's vampires have to bit the victim three days in three consecutive days, but there's room to forge your own idea of how a vampire is transformed.
Do vampires have to register with the Ministry like werewolves do? Can anyone see Hermione arguing as passionatly for vampire rights as she did for house elves and werewolves?
It's doubtful that vampires have to register unless they have magic abilities. Besides with Fenrir on Voldemort's side, I'm sure the Ministry's concern was more heavily directed towards werewolves than vampires. And no, I do not see Hermoine arguing for vampire rights because vampires drink blood, take human victims. I don't see her being an advocate for that kind of activity. At least with werewolves there is a 'cure' that's worth arguing for the werewolve sides but not vampires.
Does anyone see a possibility of a Ministry program to provide vampire with donated blood for food so they would hunt people (like the Clean Needle Program)?
Since I find it doubtful that the Ministry would care about the Ministry, I find it doubtful they would provide them with a food source/blood. Besides, in the Ministry's and wizards's minds, they are monsters because they feed off of blood so it would be hypocritical to donate blood.
We know that vampire are not sold wands, but if one came into their possesion some other way, would they be able to use it?
It depends on whether the vampire is magical or not. It's like a human: muggle or wizard. However, with their super abilities and immortality, it could be argued that a vampire is a magical being by nature, but it depends on whether vampires are divided by magic properities.
Do vampires have a place in wizard society? Would they be able to work alongside wizards, say in the Ministry, or would they be social outcasts like werewolves?
More likely outcasts. They can't stand the sun so daylight hours would limit them so unless they're security guards, I don't know how much value they would have in the workforce. They probably would be solitary, depending on other vampires. Unless their immortality/past could reveal something vitally important to the Ministry, I don't see the demand for them.
There are potions to control werewolves when they transform? Do you think there may be a potion to stop vampires from biting?
I don't find that plausible. If they don't bite, they don't drink blood. If you mean from not infecting people, well it depends on how they infect the person. But if you mean biting, I guess you could give a vampire a wine glass and tell them to drink it.... o.o Otherwise they bit they're food like we chew ours.
If you need anymore help, feel free to PM.