Wouldn't it be possible to be infected and become a vampire even if said character has already joined the dawnguard? What would happen then? Would your fellow dawnguard members attack you?

Or is it true about the rumor that the dawnguard are some form of were-beast. If that was the case, then perhaps you had to become a werebeast to join the dawnguard, which would grant you immunity from Sanguinare Vampiris.

they would likely attack you (or even cure you)
and no,the dawnguard arent were beasts (they said they werent doing a werewolf .vs. vampire thing)

I think they would either cure or kill you.

I'd be under the impression that, so long as you're fighting the Vampire Lords, you're an OK vampire. But you'd still be held in suspicion.

yes a few people have said that npcs wont attack regular vamps , only the vamp lords

I don't think thats how Skyrim vampires work, they are connected with their creator and are bound to their will...so from a RP standpoint your character wouldn't be able to turn on his kind to side with the humans or if he did he would be compelled to drain them of their blood for the glory of Clan Volkihar.

I didn't know vampirism also created a lack of free will, that's news to me.

Well the Alva chick and the vampire Laelette she created who didn't wanna do her dirty work and wanted to protect the dude Alva wanted to kills daughter but was forced too in Morthal was bound to her creator's Movarth's (Yes..the guy from the Immortal Blood book if I recall) will to try and enslave the town so they would always have a steady supply of blood and a place to hide from the sun at.

My theory as to why you aren't forced to do what your creator says in the vanilla Skyrim is because you typically kill the Vampire who infects you...also game limitations perhaps?

Also its common Vampire lore that a Vampire fledgling is linked to their creator.