Prepare to be continually disappointed. I switched my domain over when it was first announced, and although it certainly works, there are definitely some needed features lacking. Other than centralizing how emails are delivered, there isn't much of a benefit to using a custom email domain. Plus, I've yet to find an iOS app (other than Mail) that works with them.

Regarding migration, have you tried simply dragging and dropping the email from the old account onto the new one? I haven't used T-bird for this, but it works in Mail.

For migrating your best bet is probably Thunderbird or Outlook, whatever you prefer. Use that app to export a copy of your mailbox from the old mail servers, then import all of those into your single icloud mailbox using the same application.

I've never migrated mail into an icloud mailbox though so there may be some hiccups there that I'm not aware of.

1. There's an article on Apple Support about importing mail here: https://support.apple.com/en-gb/guide/icloud/mmf9877972f7/1.0/icloud/1.0

2. I think this is a Thunderbird issue. They are separate in Mail.app. It even allows you to use a catch-all rule if you're the domain owner. Maybe you could set up smart mailboxes (or whatever the Thunderbird equivalent is, I've not used it in probably 15 years)?

Thunderbird doesn't understand aliases by default (apple's mail apps on MacOS and iOS do). You'd need to add the alias under Account Settings -> Manage Identities for each alias (which is any custom email domain accounts you add, assuming you want to send mail as that user). There is only one account: the iCloud login. Everything else is treated as an alias, and doesn't create its own inbox - everything goes into the singular inbox.

As the other posts said, email won't migrate automatically. The easy way to do it, though, is setup your old email and the iCloud email in email and just drag and drop your email from the old email to your new iCloud one.