1. The 70-300 L is excellent. The non-L is very mushy at longer than 200mm - you'd be much better off cropping from your 70-200L.
2. The 28-300L is nearly a twin to the 100-400 in size, weight and handling, although it has the mount gasket that completes the weather sealing. I like it as a walkaround lens on FF. Personally, I don't find the push-pull to be a nuisance - it's fast, which is great for a lens with a broad range.
3. If you have the original version of the 70-200/2.8 IS, I would skip the extenders - it doesn't handle them well from an IQ standpoint. The MkII does very well with the 1.4x and decently with the 2x.
Assuming you have a standard zoom for FF, or will be getting the 24-105 kit, that would pair her very nicely with the 70-300 L.
I would skip the 70-300 DO. The size is great, the IQ is decent if you apply extra sharpening and contrast in post, the zoom creep is horrible, and whatever you do, don't even think about buying a brand-new copy of the lens. Used copies sell for $700-800, I bought one used and after a few months, decided I didn't like it, and sold it for the same amount I paid.