I just bought one of those lenses on your list. The Sigma 70-300 f/4-5.6 DG Macro.
But, unlike your camera, mine has in-body image stabilization by means of sensor shift. And so far I've kept it turned On. The camera is also new to me ;) Anyway, I went out today and turned the SteadyShot Off. It was a bright day today, and shutter speeds were high enough that handheld shooting was possible, but needed a few shots to get a sharp one captured. Or at least as sharp as my camera allows for.
You asked for handheld samples, so that's what I did. The camera is an entry level 16 megapixel Sony SLT A37. I did lean on a park bench, elbows on knees or whatever I could find at the beach, but the camera is in my hands (and somewhat protruding into my eyesocket) and not touching anything else.
300 mm, 1/400 sec, ISO 400, distance to target ~32 meters
300 mm, 1/800 sec, ISO 100 distance to target 12 meters
300 mm, 1/200 sec, ISO 100, distance to target 17 meters
I could not show the photos inline. Uploader got stuck. Maybe because these jpeg-photos are straight out from camera and there is possibly something that "imgur" does not get right. I did not want to ride them thru an editor of any kind.
While trying to steady the camera with that zoom pulled out to full length, I found it hard to keep my left hand so near the camera body. Focusing part is turning and the farthest away part of the lens I can hold is the zoom ring. That is not very far from camera body. Good grip on the zoom ring, no complaints, but shooting handheld with 300mm focal length is a bit on the wild side. Not even mentioning exact framing and composition. You just point and shoot, then crop the excess away later.
Focusing may be slow, I would not know if it is or is not. But the
focusing motor is definitely not silent, especially noticeable when mounted on a quiet Sony SLT camera. Edit: there is no focusing motor in this lens. Sony A37 has in-body focusing motor that does the job.