QuestMail is just a web interface (akin to gmail or hotmail, etc) to access your various email accounts as opposed to using a client software such as Outlook or Thunderbird on your PC/Mac.
So basically you'd use your email credentials you normally use on your email client on the
QuestMail website and have a way to check your email from say a library computer or somewhere where you wouldn't necessarily have access to your normal way of checking your email (phone, tablet, pc).
Unfortunately, if there are issues with the email servers QuestMail isn't really going to help anything as it's just another interface to interact with down email servers.
Hope this helps.
Brad