As a over 30year remodeler and having done countless redo's involving Tiles I would not hesitate for a second to remove and redo, don't patch a major problem.
Your going to love that porch when you get it done, sure miss mine :).
There are many plastic product's out there you can use. It really depends on where you are and what weather elements you have to endure before picking the best product. You are on the right track though. I built a square...
If you want to know then take a section of it in and have it tested but call them first, there is a protocal on how to package it up before taking to the office. Here in BC Canada they are very strick rules for handling anything with asbestos in it or even if there's a possibility. At a minimum...
I have been reading over this entire thread and the best idea yet is the iron on, your gonna have a hard time making a filler strip than skinny. The iron on is fast and easy to do.
I have never worried about it in Kitchens just make sure its at least eggshell for easier cleanup. Bathrooms however really should be mildew resistant. I have been in sooooooo many baths with water run lines all over the walls because lack of putting the fan on, at least this way your covered