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> I cannot figure out how to move my old emails and contacts from the Thunderbird to the G Suite account.
If you talking about the contacts of the specific email account of TB (not old email service), you may simply export the contacts in CSV format from TB and import it to GS. Manual modification might needed if there are a few differences in CSV specification.
> I wonder if the fact that the old account and the new account have the same address is also creating confusion.
The email address is kept as you own the domain and it is not changed. What has been changed was its email server's configuration (both hostname/IP address and protocols/ports).
> I understand that the name on G Suite can be changed and then changed back but I'm not sure if that's necessary.
What name are you referring to? What do you mean by "can be changed and then changed back"?
Thanks for your message. But it seems I can't be available during this weekend as all have been arranged. I am really sorry about it. Can we set a time on weekdays?