John, thanks for your reply and the information you provided. Unless my technique improves, I'm reluctant to remove another row of tiles. I'm afraid the chipping will continue until no more tiles are left at all, and that's a lot of tile!
Do you have any suggestions for a method of getting the tiles off clean? I've been using a Dremel with a grout removal bit to get the grout out, although I found it difficult to "steer" — it had a tendency to move off the grout and onto the softer tile material. I have also bought a manual carbide saw for grout removal, but haven't tried it yet. I can only guess that my difficulties may be in part caused by, as you have suggested, the fact that the grout I'm trying to remove is portland cement.
I'm still not sure on how to deal with the uneven surface. What should I use to even it out before putting on the new tiles? I want to be sure there is good adhesion to the existing mortar substrate.
Thanks again!
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Arthur