Install Windows 7 on Toshiba Satellite L75-B7150

Download and run the Windows 7 Upgrade Advisor to see if your PC is ready for Windows 7
Link:
https://www.microsoft.com/en-us/download/details.aspx?id=20

Is it now Windows 8?  You can upgrade to Windows 10 for free
You can run a dual boot system with the original OS and Win 7
Or Install Win 7 as a virtual OS

Not sure why you want Win 7?
To run old programs?
Refer to
"How to Make Old Programs Work on Windows 10"
http://www.howtogeek.com/228689/how-to-make-old-programs-work-on-windows-10/

You can also make 10 look and feel like 7
Refer to
"How to Make Windows 10 Look and Feel Like Windows 7"
http://www.laptopmag.com/articles/make-windows-10-like-windows-7

HD failed and I have a W7 DVD. Plan to immediately move to 10

Should install OK, but Japanese PC makers use their own drivers and not the generic drivers of MS.
Win 7 covers most drivers, so as long as the NIC drivers work you will be able to upgade to Win 10 and update drivers online.
You can only try and If you strike driver issues, try to find them on their site, or contact Toshiba support.
I have wiped Japanese OS PCs and put Eng. OS on Toshiba PCs and had to get drivers from their Japanese site for Japan only models ( I live in Tokyo )
Your model is Eng. so it has Eng. driver support.
Link:
http://support.toshiba.com/drivers

here a detailed view of the fresh install :
http://www.wikihow.com/Install-Windows-7-(Beginners)

You will still need a Windows 7 license and activate that before you can upgrade to Windows 10. It would be much cheaper to use the recovery media of that PC to restore it to factory defaults, and after that do the upgrade to Windows 10, or order a set of recovery media for your PC from Toshiba if you failed to create them yourself, or lost them. Costs for recovery media are usually far lower than original installation media with retail license keys.

I have a W7 license.  Issue is getting the PC to boot from DVD.  The screen is broke and 25% isn't visible.  Not sure what keys to hit.  What about UEFI?

Connect it to an external display. If you use the 64 bit version of Windows and the DVD is original, UEFI is no problem.

Tried connecting to an external... guess that the external should reflect the computer straight away?  Right from boot?  Appears that this laptop has bigger issues....

Normally it should show the same things on the external display. Sometimes you also need to use a special function key to change to external display, like F5, F7 or F8. Normally the key has the appropriate symbol on it...

Given that I cannot get this to boot to a DVD and cannot see via an external, I've got to assume that this thing is damaged beyond a cracked screen.  Should probably give it up.

you can check if the video plug came loose, or any other parts
in such a case - i disassemble the board from the case, and check all parts on the bench

Yep