Okay, so I decided to play this character http://s1287.photobucket.com/user/quinbus777/media/ScreenShot2269_zpse513a007.png.html after seeing the post about Asian races in Skyrim and I made her just to see if I could.

Since I also decided the Akivri/Blade gear and questline were appropriate for her backstory I decided to play her.

Then it occurred to me. When you help the Blades on the MQ and you recruit and, say, recruit members, and/or kill Parthurnax and what not, are you automatically a Blade?

You don't take an oath, or have a ritual, or even say to Delphine, 'Sure I'll join". Does being Dragonborn make you one automatically? Do you have to kill P only? Do you have to kill P and recruit? At what point, if ever does it become official. All the other questlines have a clear demarcation, an oath, a ritual, or a 'here ya go Archmage have the robes of office' sort of thing.

As a side note: I know I don't have to be in the Legion, but would a Blade definitely want to join the Legion, or stay unencumbered by that obligation?

You never become a member, only a tool to Delphine. She's controls you now.

I don't think a Blade would join the Stormcloaks. You can if you want, but they have mainly been associated with the Empire and Dragonborn Emperors...which no longer exist. They did fight in the shadows though.

The Dragonborn is not a member of the organization that fronts itself as the Blades (Pardon if this is not palatable but there has been some discussion on this). Delphine's faction claims to be there to guide and guard the Dragonborn but that has been hotly debated as well and FWIW I stand on the side of those who don't believe her.

Should you side with them killing P seems like the only logical choice. This does not make you a member either though it does give you access to a sword and armor for that faction.

Being on either side of the MQ I see as a non issue since Delphine's faction desires only to slay dragons and seems to have no interest in the civil war, you could just as easily do Season Unending.