The main thing is you do not need to transfer your domain, you can buy email service and change the DNS records on the domain to point to the new service.

My #1 advice is to keep domain and mail/whatever else hosting separate. You can transfer your domain to another registrar, and then get an email hosting service that allows you to use your own domain.

That way you can move your email to another provider without also having to move your domain and vice versa.

My domain registrar is INWX, and I host my mail server on my own VPS so I can’t speak to the quality of any mail service but Hostinger allows you use an external domain.

That DNSSEC status does not have anything to do with being able to transfer your domain AFAIK, that is instead generally something different called Transfer Lock.

To transfer your mails, what I did in the past was just connect the two mailboxes via IMAP to a local client and copy everything from the old mailbox to the new one (or to a local one first, whatever). As long as both sides support IMAP, you don’t have to have any special support from either provider. But it’s probably nice to have.

You can connect non-Gmail mailboxes to the Gmail app but there are better alternatives. Thunderbird as you said, for Android there’s K9-Mail. Personally I use KDE’s KMail and Apple’s Mail app on my computers/phone. YMMV.

if you want just a custom domain tuta or proton allows you to but if you want to fully host on your vps look at somthing like moxmail as a backend and a vps in countries that are not required to turn over data so to get full control find a vps that accepts crypto like Monero then install your mail provider of choice like moxmail and to access i think moxmail has a web pannel you can access or you can use a mail client like thunderbird youd just need to input your smtp imap stuff wich can all be found during setup also make sure your host supports email like some wont let you open ports for email