Some of the Windows 10 installation issues boil down to hardware compatibility. Some SSDs, for example, aren't 100% compatible with some UEFI implementations. Ditto for flash drives and certain versions of Windows installers (this was a major source of headaches for the early X79 and X99 motherboards). Rufus helps with making sure that the image is bootable, but if you can't see the SSD drive in the partition window, then there's an issue with either the drive, or the UEFI BIOS.
Some of the things you can try include:
1) Setting the UEFI restriction to "Legacy" or turning it off.
2) Ensuring that the drive is set to AHCI mode. RAID can also work, and will also pass along TRIM commands.
3) Plugging the drive into a SATA port connected to the chipset, or
4) Plugging the drive into a SATA port connected to the third-party controller.
5) Try a different DVD drive or flash drive.
6) Updating the motherboard's BIOS if there's a new version available.
7) Plugging the SSD into another PC and updating the firmware.
8) Installing Windows from one of those "All in One" discs.
9) Install your copy of Windows to the drive from another PC, but don't activate it, install any drivers, or put in a product key (
see here).
#9 is the second-to-last resort, since it typically works, but may produce errors since the UEFI implementation may not be the best, or may differ from board to board. If you use an older, non-UEFI system to install Windows to the drive, say a LGA755 platform, you'll have to switch to using the legacy BIOS mode as well.
As a last resort, change the drive and/or motherboard.