Yeah, autofocus is done with the lens wide open. I've just checked and confirmed that, for example, my 40D has horizontal and vertical sensors that can AF at f/5.6 and a central precision cross-type sensor that requires f/2.8 or wider. The pro bodies also have sensors that can AF at f/8 or faster, but the consumer and prosumer lines don't (the 7D may be different).
It's likely that the same is true with Sonys.
I can't help with the issue with the Tamron as I have no experience of it.
There are people having similar experiences (though not with the same kit), and a couple of suggestions of possible causes here:
http://photo.net/nikon-camera-forum/00X0DQ