Provided you connect to your email server with IMAP then you can use any email client or webmail and you'll see the same email in the same folders - including sent mail. I use Thunderbird on several Linux and Windows PCs and Profimail on Android. If necessary I use the webmail program my provider (1&1) provides but I find it slow and inconvenient.

I used to use POP with several devices but IMAP is much easier. As for busting storage limits it's really down to discipline. In Thunderbird I often detach pictures or large files leaving the emails behind. I can also periodically archive the email onto a PC.

For sending large files I use dropbox-for-filelink* Maybe your employees could be encouraged to use something similar.

*Despite the comments on that page I find it works fine.