Big name websites generally have enough users that email "just works". Smaller websites are more likely to use misguided measures such as a bad email validating regex (hello to anyone with a non-standard TLD!), only allowing gmail, or blacklisting domains like these.
From my personal experience it is best to have a secondary email account on a provider that is usually not blocked (like gmail), to keep your primary email account clean.
The only correct to validate email addresses is to just send a message there and see if the user can click the confirmation link.
Chances are that would be the next step in any signup flow anyway, so why introduce this artificial middle step of "validating the email address"?
Ever heard of Magento? They have that built in, at least in version 1. But it's a fixed list with "valid TLDs", anything not on that is not accepted when registering.
Feels strange, when you can't register on your own shop...