Indicating work was performed as a student for alumni
assuming I use the gmail address, should I add a footnote to my name saying "Work completed while an MSc student at institution" to justify the affiliation?


No, because you never need to justify right in the paper why your affiliation is indicated the way it is, or why you picked one e-mail address over another.


  I've seen this done for papers published by students about work done during internships at companies, such as "This work was completed while name was at company"


I can imagine the benefit here is to inform readers that company is not a likely place to get in touch with name any more. That can be useful information, but it is most likely not provided just as an explanation of why a particular e-mail address was chosen.