Publishing master's research while in a different PhD program
It's fine.  Similar things will happen after the Ph.D., after job changes, etc.  Your current advisor doesn't need to be listed or involved in the process for work done elsewhere.
It is frequent (but unfortunate) that people write the thesis first and then produce articles.  However, I think it is better to be publishing along the way.  And this is not unusual at all either--so you are a little wrong that it is never done.  All that said, converting thesis chapters into papers is very normal and fine.
List the previous institution first with a footnote or parenthetical for current affiliation and contact email.  Do not list the old email.