Dead HDD? try to recover data or destroy it?

I would recommend just running it in one go because every time you read from a failing drive you are potentially causing it to fail more and you don’t know if you will be able to continue again if you stop it.

You can add status=progress to monitor as it runs. It will take a long time, yes.

You can compress the image by having dd write to stdout and piping to a compression utility as you run it, but there’s no way to know how much space that will save you in advance so you should still write to a >=4TB drive IMO just in case and I’m not sure it’s really worth dealing with the compression at that point.

Keep in mind anything you do at this point is at your own risk. Any interactions with the drive may make the issues worse and make any remaining data completely unrecoverable even to a lab.