Shutter speed should be at least 1/focal length or faster where at all possible unless panning of course. And because it's a biggish lens, even at 120mm, I would think of 1/250th as the slowest shutter unless forced into to something slower.
Monopod is useful in some situations, tripod for others.
Depends on what you want to shoot, but I put a lens protection jacket on my non OS 300mm on the odd wall or fence without damaging the lens (or the wall)
Stopping down helps with sharpness, but hurts shutter speed, so your choice..