I am doing a full install of 16.04 to an external 300gb USB HDD. I have followed all the instructions (booting from live media, then using the install option, taking care to setting the grub to point at the external usb drive (in my case sdc1).The installation proceeds normally however the grub is written to the internal HDD drive. Fortunately I know how to recover the grub and to boot normally. But something is wrong. I don't want to open my laptop to unplug the internal HDD. Any suggestions?
I faced the same problem with my Lenovo ideaPad Z50-75 Win10 OS. I was able to do the installation on my external USB HDD after disabling the secure boot and enabling legacy system from BIOS. I also installed the boot loader on the same external USB HDD. Updating Ubuntu sw (after installation) messed up the root system, which was easily fixed with initramfs fsck-command. However, if changing the boot order in BIOS the system could not read the Win OS HDD (a SATA drive in my case). The other problem was that I could not connect to internet via wifi (propably something to do with Broadcom driver and USB HDD connection). The wired connection worked fine. One more problem was that the Ubuntu boot loader could not start Win OS (pointing to wrong address ...). Removing the installation is no problem: you can delete the ubunto boot loader from BIOS for example with easyUEFI (free sw). After that you can return the BIOS settings to original values and everything should work fine with Win OS.