The answer to your first question is pretty straight forward -- rule of thumb is that to avoid camera shake, use a shutter speed that's the reciprocal of your full frame equivalent focal length. e.g. when zoomed all the way out (38mm), use a shutter speed that's 1/38 or faster. So basically, 1/40 - 1/1000. When zoomed all the way in (380mm), use a shutter speed that's 1/380 or faster. If you need to use a shutter speed slower than the guideline recommends, then yes, image stabilization may be helpful.
The answer to your second question isn't as clear. It would be very helpful if you could post a few example images with EXIF data, but you could also try using a slower shutter speed (and image stabilization, if necessary) and/or higher ISO.