After verifying site ownership, is it ok to remove the <meta name="google-site-verification"> tag?
No. If you want to remove it and remain verified, you'll need to use another form of verification.
If their email address is on the domain, use https://www.icann.org/users/password/new to have the EPP code sent to them. Transfer it to a new registrar and you now have control of nameservers and thus DNS.
If you do as Trebor suggested and grab all the email, then you can migrate the WordPress site using a plugin (I like all-in-one migration [no ...