Hi
Your question leave a lot of questions to be asked before a sensible answer can be given!
Did this start happening after a system update? If so which one.
What type of email account is it? gmail, hotmail, yahoo, pop, imap.... Which app do you use to receive emails, gmail, email etc.
Its unlikely that deleting emails will delete contacts. Its unlikely that your S5 is full up with emails (although not entirely impossible!). Depending on the type of account, deleting emails from your phone may delete them permanently everywhere so that they are lost along with any attachments.
Having said all that, have you tried opening your email app, going into the app settings and enabling auto syncing and notifications?
Roger