There's an offline gmail chrome app that lets you work that way. Also, turn on two-factor for them. They can receive the number via SMS, and it'll help prevent them from being phished. Once set up, it's easy to understand how to do it, and they only need do it every month. (There are a few email providers that provide 2-factor).
gmail can check a pop3 account on your behalf, and you can set your 'from' address (I haven't checked the constraints on what you can set it to...). So there's not necessarily a need to change email addresses to use gmail.
if gmail is blocked, then you're in an unusual situation where nobody here can give you good advice without knowing more about what's going on.
I'm advocating gmail here for three reasons:
(1) Really good spam filters and phishing warnings that can help keep out scams
(2) Two-factor authentication
(3) Easy setup with a chromebook.
With the last, they can keep all their stuff on drive (and you can just log into drive to help them), and you can chromote in to see their desktop and help. Even video-chat while chromoting.