The general rule is that once a line is used as a drain, it cannot be a vent. To keep a vent a vent, you need to connect one from below high enough so it can't inadvertently become a drain...so, code requires it to be at least 6" above the highest fixture. So, no, that isn't the fixtures in the basement...the connection point for a vent must be above the highest thing above it. Depending on how the vents are run in your house, you might have one for the first floor stuff, and a second one for the second floor stuff, or they both combine into one before going through the roof. You could make one for the basement stuff, or, combine them in an approved manner. Basically, if the first floor is properly vented, you'd only need to go the 6" above that floor's highest fixture flood plane (or 42"), since a vent is a vent. In an older house, though, they didn't always do it such that it would pass codes today, so you might have to go higher to make the new stuff pass.
AAV's are a mechanical air vent - they let air in, not out. They are not always allowed, and even if allowed, the inspector may give you hassles if there's a way to do it 'right' (i.e., without one). depending on the layout, an AAV may or may not be acceptable. Where ever you do install one, though, it must be accessable, not hidden in a wall without an access plate, since it will eventually fail and need to be replaced.