My distribution certificate is expiring on June 7th, along with all of my provisioning files. How do I properly renew it? Should I revoke it now and request a new one? If I do that than will all my live apps be taken down?
Your live apps will not be taken down. Nothing will happen to anything that is live in the app store.
Once they formally expire, the only thing that will be impacted is your ability to sign code (and thus make new builds and provide updates).
Regarding your distribution certificate, once it expires, it simply disappears from the ‘Certificates, Identifier & Profiles’ section of Member Center. If you want to renew it before it expires, revoke the current certificate and you will get a button to request a new one.
Regarding the provisioning profile, don't worry about it before expiration, just keep using it. It's easy enough to just renew it once it expires.
The peace of mind is that nothing will happen to your live app in the store.