Once the emails are fwd-ed then a "copy" of those emails will be on gmail or hotmail. It should not matter which provider you will choose, each of them have good things or not so good things. So it's up to you. Please note that fwd-ing will not apply to old emails but only to the new emails received after you have set that fwd-ing rule.

When your service provider will delete your email account: -nobody will can send an email to that email address (as it will be deleted - and the sender will receive an error that the email address is wrong). -you will can not access the mails from that server (but you will can still see those copies from gmail or hotmail) -any client email software (like Thunderbird) will not be able to connect to that server for that email address -so that email address will be gone (will not exist anymore)

The emails from Thunderbird will remain in Thunderbird if the account connection was set as pop. If the connection was set as imap then, after your email will be deleted, you might not be able to see anymore some attachments/texts - you can prevent this by copy all your old emails (in TB) to a "local folder".

Thank you very much for your reply, svlad2009! Very informative!

Clarifications: - I have set a forwarding rule for emails incoming to my soon deleted email account. -I manually forward older emails from my soon deleted email account to my Gmail. Do any of these clarifications affect your advice?

Again, thanks in advance for any info!

Well, if all emails are already fwd-ed (or will be automatically fwd-ed) to gmail then all those emails you have them on gmail. Everything you have on gmail will not be deleted when your service provider will delete that email address. If TB is connected to gmail then also those emails will be visible (w/o any issues) in TB.

Thank you very much again, svlad2009!