How can I avoid this in the future?
Easy - by using the builds from the download page. There are no focus dots on the 5D Mark III, that I know of.
However, there are some low-level settings (lv_af_raw) that cause some kind of subtle focus dots. If these were changed in the build you were using, then... sorry, it wasn't me who introduced this problem

The pattern looks quite interesting though - I've never tried to map it. These dots are very likely introduced by Canon software, but what exactly they are used for, I have no idea.
What is the best way to remove Focus Dots?
I don't know what is the *best* way (that's an optimization problem), but the dots should not be there in the first place - as explained earlier. Of course, since they are in your clip, you can use the FPM method to fix them - it will work, but there might be better ways. Anyway - I've noticed one of dots at (1406,915), but I could not find it in the FPM file. IIRC, the coords are relative to the top-left pixel on the sensor, not in the recorded image (since one map needs to cover all possible resolutions from a particular video mode), and since I don't remember the exact details... let's reverse engineer the offset. By comparing the DNG with and without that pixel in the FPM file, I've noticed a difference at (1254,783). So, to interpolate our bad pixel, you'd have to write "1558 1047" in the FPM. I could not find it in your file, but I could confirm that, by adding it, that particular dot disappears.
If you already mapped your bad pixels - in the FPM file - you know what you have to do: shift them into the right place. I don't know which of the dots from your list is the one I've identified - worst case, you have only 295 possibilities to try

Good luck!