Can I do a brand new time machine backup on a new drive but still be able to restore from old drive, old backup if needed?

Make sure the machine storage isn't too full to begin with. If the computer is over 85% full, it will take that much longer.

Limit its access to internet while trying to back it up.

Also make sure the hard drive you are backing up to is formatted appropriately. A backup to an ExFAT or FAT32 drive is going to be a lot slower.

https://discussions.apple.com/docs/DOC-3003

As for reading the old backup, it is best to do it when no other drive is connected.

The other thing to keep in mind, the capacity of the backup drive should ideally be at least twice that of the boot drive it is backing up for archival purposes.

If Time Machine is turned off, when you attach the other drive, you can manually search for files by name within the backup and pull out important files with Devon Technologies EasyFind

https://www.devontechnologies.com/apps/freeware

Do not mess with the file or folder structure in any Time Machine backup drive, as that's important for Time Machine to recover information through its own interface.