Where do I find the authorization code?
In the current registrar's control panel, usually beside the domain's own settings, or by email on request. It is sometimes called the EPP code or the transfer key, and it expires, so start the transfer soon after you get it.
Will my website or email go down during the transfer?
Not because of the transfer. It changes the registration, not the DNS records: as long as the nameservers keep pointing where they point now, visitors and mail notice nothing.
Why would a transfer be refused?
Almost always one of three things: the domain is still locked, the code is wrong or has expired, or the name was registered or transferred less than 60 days ago — registry rules block a move inside that window.
Can I transfer a domain that expires next week?
Renew it at the current registrar first. A transfer takes a few days, and a name that expires in the middle of one can be suspended while it is in flight.