I have this lens and I have used it with my ancient 1D mark II with the Canon 1.4x teleconverter and auto focus is surprisingly good in good light and with the 8.2mp sensor images looked about the same. I've also used it with the Canon 2.0x teleconverter and of course at f/11 (2xf/5.6) you do not get auto focus and the image quality is severely degraded but it does work if you don't mind manual focus. My teleconverters are the original or 1st version t/c's and your results may be better with the mark II or mark III t/c. With my 6D you do not have auto focus with a t/c, for some reason Canon crippled that feature while it's always been available with my Nikon bodies and lenses. I purchased a 5Ds a few ago and haven't had a chance to test the 100-400 with a t/c but Canon claims this body does support f/8 auto focus. My plans are to sell it in the near future and pick up the Canon RP because it's cheap and video is greatly improved over both my 6D and 5Ds.
Maybe a 1.4x, but not a 2x, and only on a full-frame. This will degrade the image quality even more, and especially when using an APS-C camera, an extender degrades the recorded IQ of the lens even more.
What you mean is the reciprocal rule: without an image stabilizer (which this lens has), your shutter speed should be 1/focal length. Nothing to double. When you use an APS-C camera, you need a short shutter speed. But with a good IS, you can expose for much longer. When properly exposed, the images should have the same sharpness.
Tamron has some nice zooms at reasonable prices. Check DXO for lens ratings etc. You will most likely want a zoom not a long prime lens. Make sure the lens is appropriate for your sensor size. You can save money buying used, it is not like glass wears out. Get one with built in stabilization, good stabilization is very important on long lenses.
I much much much prefer version 2 of this lens it has much better stabilization and sharpness. If I did not have the money for version 2 I would look to a third party rather than the original 100-400 Canon. The original needed replacement.