I have a website that is currently hosted at a company that also hosts my email address. I am switching hosts soon, and cannot have any interruptions in emails. What is the best way to go about this switch without losing any emails that could possibly come in during the switch (DNS propogation?).
Your best bet is to setup a secondary MX record and server to catch all of those e-mails during the migration.
There are lots of companies that offer Backup or Secondary MX record hosting services, some free, some not. Either way, the entire purpose is to sit there and get mail and then deliver it on to the primary MX box when its available.
It's ideal for your situation and should deliver e-mails to your new host as soon as it is online and available. Its still best to migrate stuff like this over the weekend or at night when traffic volume is low.