Dual boot menu with Ubuntu and Windows 8 not showing up

Solution 1:

According to your description, you have correctly installed the GRUB bootloader in UEFI mode. You do not need any BIOS-boot partition, nor any other partition change.

The only remaining issue is to set up your BIOS (UEFI firmware) so that it boots by default the GRUB entry instead of the Windows entry. In other words, we need to find a way so that you won't need to press 'Esc' then F9 to reach the GRUB menu.

To perform this, please:

1. check the 'Boot order' menu in your BIOS, and try to place the "Ubuntu" entry first. Then save the change and reboot. If GRUB still does not appear by default, please follow step 2 below.
2. run Boot-Repair --> Advanced Options --> tick the Backup and rename the Windows EFI files option --> Apply. Tell us the new URL that will appear, reboot, and tell us what you observe. The PC should now boot directly onto the GRUB menu.

Solution 2:

You have UEFI and have grub installed to efi partition. Your system did promote the flash drive to sda and made your 1TB drive sdb, but it still should boot ok.

You must be booting Boot-Repair in BIOS mode if a grub reinstall is asking for a bios_grub partition. That is only required to correctly install grub to the protective MBR that gpt partitioned drives have to boot in BIOS mode. But you really want to boot in UEFI mode.

I have not seen boot issue on any UEFI based system with grub or kernel files far into hard drive. But some BIOS based systems would not boot if kernel or grub were beyond 100GB on drive. Those needed a separate /boot nearer start of drive.

From UEFI menu, can you boot Ubuntu with secure boot on or with it off? It shows you have also installed signed kernel so it should even boot with secure boot on.

If you then still have issues after grub menu, then it may be something else like video issue. What video card/chip do you have?