"Error: Invalid signature" when configuring dual boot Ubuntu 14.04 and Windows 10

There are some peculiarities about your installation:

• Your computer has two Windows Recovery Environment partitions, /dev/sda1 and /dev/sda5. My guess is one is from your previous Windows installation and the other is for the new installation, but I can't be positive of that. I doubt if this duplication is causing you any problems per se; I mention it simply because it's a sign of a history of changes to the machine, and such changes may be both the cause of problems and make it difficult to figure out what's wrong.
• Your EFI System Partition (ESP) is /dev/sda4. This is perfectly legal, but is unusual; the ESP is normally /dev/sda1 or /dev/sda2. The ESP holds EFI-mode boot loaders and related files. My hunch is that the ESP was created late in the machine's history -- perhaps it was Ubuntu-only at one point and then you installed Windows, which created the ESP in this unusual location.
• The Boot Info Script output doesn't identify any EFI boot loaders on the ESP. My suspicion is that they do exist, but the script has omitted these entries. (I've seen this happen from time to time.) OTOH, if those files are missing, then it means that Windows has failed to install its boot loader. The same may be true of Ubuntu, too; but see below....
• Your disk uses the GUID Partition Table (GPT), which means that Windows will install to and boot from this disk only in EFI mode; but there's evidence of a BIOS-mode installation of GRUB, the standard Ubuntu boot loader. Such a mish-mash means that you'd not be able to boot Windows from a BIOS-mode GRUB on this computer. It could be you've also got an EFI-mode GRUB and had been booting that way; or maybe you had been switching between OSes by using the computer's built-in boot manager (typically accessed by hitting Esc, Enter, or a function key at start time).

At this point, your best bet is to install an EFI-mode boot loader for Linux. There are two ways to do this that are relatively safe and easy. Both work best if you first enter your firmware and completely disable any option to boot using BIOS, CSM, or legacy mode. (Those are three terms for the same thing.) Sometimes these options are phrased the other way -- you must enable EFI or UEFI support. With that done, do one of two things:

• Method 1: Boot your Ubuntu install disk, ensure you've booted in EFI mode by looking for the /sys/firmware/efi directory (if it's present, you've booted in EFI mode), install the boot-repair package, and run it. This should repair your installation by installing an EFI version of GRUB. This usually works, but occasionally it doesn't. I realize you've already run Boot Repair, but my suspicion is that you ran it in BIOS mode, which of course means it would install the BIOS-mode GRUB, which will be useless for booting Windows.
• Method 2: If you haven't done so already, disable Secure Boot in your firmware. You can then download and prepare a boot medium from the CD-R or USB flash drive version of my rEFInd boot manager. Boot to it; it should show you a menu with at least one option for each of Windows and Ubuntu. Test these to be sure you can boot both OSes, then boot Ubuntu. At this point, you can install the rEFInd Debian package or PPA. (Alternatively, you could install the EFI version of GRUB.) You may be able to re-enable Secure Boot at this point, but you may need to jump through some extra hoops to get it to work.

There should be no need to re-install Ubuntu, and probably not to re-install Windows. BIOS-mode and EFI-mode installations of Ubuntu are identical except for the boot loader (and related partitions -- but you've got the partitions to handle both modes).