Fortunately, Yahoo now provides free forwarding as per the following:
Mouse over the Settings menu icon| select Settings.
Click Accounts in the left pane.
Click Edit beside "Yahoo account."
Enter the address you want to forward your email to and select:
Store and forward - Email is saved to your account and sent to the forwarding address.
BlueHost already have a Knowledgebase Article about How to access your e-mail with Gmail. You can skip the Outgoing Mail (SMTP) section as you presumably won't be using their service to send mail any more.
These are basically the instructions for using Google's Mail Fetcher to pull in old email, but they include BlueHost-specific settings. BlueHost had ...
The issue is that gmail is linking the two messages in a conversation.
To only delete one of the messages you will have to:
Select the conversation
Open the message you want to delete.
Click the three vertical dots on the right hand side of the message.
Select "delete this message".
You will then see a popup at the bottom of the screen telling you the ...
I would think that the access was forbidden - temporarily or permanently, for unknown reason; not necessarily the mails are gone. You are just forbidden to see them.
Contact them why you are locked, or simply try again after some hours.
And maybe make a backup, when you have access again.
Without more details, I think that the best option is to do the bulk operations in http://mail.google.com just before you could left alone your Thunderbird instance for several hours, as the sync process could take "a while" and consume "a lot" of resources of your computer.
How actions sync in IMAP - Gmail Help
Deleting IMAP messages - Gmail ...