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1. Do I need to roughen the surface of the old tile? I've read a belt or oscillating sander will work, not sure what type of sandpaper. Is this true and necessary?
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Yup! Ruin it as best you can. A silicon-carbide abrasive sand paper would be best and easiest if you can find it. Otherwise aluminum-oxide sandpaper will work. Use about a 60 or 80 grit.
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2. Should I place a waterproof membrane between the old and new tiles? If so is a liquid or rollout under-lament preferred? Specific brands would helpful.
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Why?
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3. What type of thinset? Does this depend on if a waterproof membrane is necessary? I'll be shopping at HD or Lowes so any brands you recommend that I can find easily would be a big help.
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This will require a quality modified thinset. Something better than the entry-level products they sell.

If your scuffing of the old tile is successful (no shiny glaze remaining) Home Depots Versabond should work okay for you.