Can I install Ubuntu on an exFAT hard drive?
No, you can't install Ubuntu on an exFAT partition. Linux does not support the exFAT partition type yet. And even when Linux does support exFAT, you still won't be able to install Ubuntu on an exFAT partition, because exFAT does not support UNIX file permissions.
I would use Gparted to create, move and resize partitions, as it supports many more file systems than Windows' admin utility. Gparted can move and copy exFAT partitions, but it can't create, grow or shrink them.