1.What partition is Ubuntu's root path on?

Check your drive partition by running the following command ls (hd0,1). In my situation, I found the partition which has total size 1000204576KiB(approx. 1TB).

grub> ls (hd0,1)
grub> ls (hd0,2)
grub> ls (hd0,3)
        .
        .
        .
grub> ls (hd1,1)
grub> ls (hd1,2)
        .
        .
        .

If your drive partition locate (hd1,2), run the following command and set root path.

grub> set root=(hd1,2)

2. What is the path to the kernel image, the drive name, and initial RAM disk path?

If your drive partition locate (hd1,2), run the following command.

grub> ls (hd1,2)/boot

And, please look for [kernel image path] root=[drive name] and [initial RAM disk path].
In my situation, I run ls (hd1,2)/boot and found above information from the bottom line.

grub> linuxefi /boot/vmlinuz-2.19.0-35-generic root=/dev/sda2
grub> initrdefi /boot/initrd.img-2.19.0-35-generic
grub> boot

Reference

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