The 70-300 DO seems a good lens. I haven't used it myself, but a friend of mine does and she's happy with it, gets good shots with it.
Of the lenses you mention, I'd definitely choose the 70-200/4 IS. I just got one as a backup to my 70-200/2.8 IS and am finding it an excellent alternative. It's a real workhorse of a lens, like all the Canon 70-200s.
You can use the 70-200/4 IS with a 1.4X teleconverter. With very few exceptions I wouldn't recommend a 2X on it or any other zoom, since image quality will take a serious hit and most will not be able to autofocus, except with f2.8 zooms or using the lens on a 5DIII or 1D series camera. I have 70-200/2.8 "Mk I" and 2X Mark II, and will not use the combo... to much loss of IQ (I use the 2X on a couple other lenses). One exception regarding IQ is the latest 70-200/2.8 IS II and the 2X III, which seems to work pretty well together. Some people seem happy with it, others have not been. Of course, it's a very expensive and rather large combo.
All the 70-200 lenses are Internal Focusing (IF) and internal zooming. This means they don't change length when zoomed or focused, making them better balanced and more easily sealed against dust and moisture intrusion.
All the 70-300 lenses are not IF or interal zooming.... this allows them to be smaller when retracted all the way. Variable aperture designs (i.e., f3.5-5.6 instead of just f4 designation) also allow lenses to be smaller.