I can't say what is or is not code in your area, but around here framing in a basement is pretty close to anything goes. There are no load bearing walls or anything like that, no load from above, so headers, etc are unnecessary from a technical aspect. That said, it is probably best to frame everything "conventionally", just don't sweat it if you have to do some funky stuff to get around ductwork etc... Not sure what you mean by a bottom plate in the door opening tho... That would be a tripping hazard.
Ultimately, you want to know what your inspector is looking for, so just give him a call and ask.