Let's say I have two 7' rods, both from the same line from the same manufacturer (just to normalize things).

Both are rated for 1/4 to 1 oz lures. One has an extra-fast action, the other has a medium action.

Which rod will cast a 1/2 oz lure farther? Which one will cast a 1/4 oz lure farther? (the line, reel remaining the same)

The faster action rod will be "stiffer", or have more backbone. So it would most likely throw the heavier bait further, everything else being equal. It's the opposite for the smaller bait. The medium rod will probably be more versatile, but you'll likely lose some bottom "feel" or sensitivity with it compared to the faster/stiffer one.

I disagree.  The medium rod will cast farther because you are loading more of the rod during the cast and can transfer more energy to the bait.

   

Depends on who is holding the rod.

Who is holding the rod, and what kind of cast is being made will decide the issue. The XF will load up much faster allowing for longer, easier, and more accurate pitches and in close work. The slower rod may give a little more distance on a full overhead cast. Just my experience, your mileage may vary.

i cast further with a slower action than ex-fast.

The slower action will load more and transfer more energy into the cast, resulting in a longer cast.  However, there seems to be a point where the action causes the tip to be too "whippy" and you loose power on the cast when the bait weight increases.  I had a G-loomis CBR843 for a little while and felt like the action hindered the power and had trouble casting baits in the upper range of the rating for the rod