I already know this, it is already done. I don't want to run the game directly by the executable or by a shortcut (however for now I must do this if I want to play), but by Galaxy.
Galaxy don't care about the shortcuts, it has its own proprietary system of properties, in some files the gamer is not supposed to modify (it is not documented, for security I guess). It seems I know better Galaxy than you :)

Maybe I didn't understand JI's post, but it seems to me that he told you how to set the *game executable* properties (not the Galaxy shortcut.) That has nothing to do with a shortcut--you are setting the executable file that runs the game directly. When you run the Galaxy front end, it runs a Galaxy program that in turn runs the game by way of running the *game executable*--if the properties are set on the game executable they will run fine whether you run the game through Galaxy or the standard non-Galaxy shortcut, because the properties are not set in either shortcut--but directly in the game executable.

Setting the exe or its shortcut, it's the same. Set a shortcut, then see the exe: it has taken the shortcut options. The reverse is true too.
Once the exe is set, like you I prefer to launch its shortcut rather than always search for the exe in folders. So, I set the exe, and as it still does not work by launching Galaxy then I use the shortcut of the game. No, it's not the same, Galaxy does not understand the propeties well. It's not direct, Galaxy reinterpreter them. Often it works, sometimes not (like for UFO:Aftershock, Galaxy does not understand all the options).
If it was the same, yes TQ would run fine through Galaxy (TQ is fine by its direct shortcut, not by Galaxy).

OK, fair enough...I don't use Galaxy...for this reason and several others...;) Galaxy isn't needed for any of my GOG games and it usually just gets in the way. Not passing through Windows properties for game executables sure seems like a major bug, to me--BTW, I'm running Win10x64 build 14393.xxx and the game runs fine for me with mouse control full screen--not a problem at all. Runs like a top with no compatibility modes set so it's running native under Win10x64. (GOG-4)

It's kind of funny...I bought the game today and as I thought this game was really a masterpiece with the fan patch for the original version of TQ:IT, I was excited about picking up the new and improved Anniversary version for <$5...until I pulled up my Steam account (where I had bought the original TQ/IT versions only to find that Steam had already added it to my account for free! Heh...;) So now I have two versions--I don't mind--it's actually sort of amusing! I'm still at the very beginning--haven't killed the first major bad guy, yet...but I've been comparing this Anniversary edition graphics to my original fan patched version 1.17a(?) I think. Kind of surprised not to see much difference...really having to look for it to find it--and then I am not sure..! I can run the original game at the same resolutions as the Anniversary game...maybe the differences will become more obvious as I play...!

I must at least put Win 8 compatibility (XP3 and Admin might not be really needed nowadays),
maybe my mouse is too old (Saitek Cyborg v3)?

Yes, it's so hard to tell sometimes as it could be a number of things. Even something to do with the GPU and hardware cursor support ijn the drivers, etc. Forgot to add that my GPU is an AMD R9 380 4GB, running the 16.8.3 Crimsons--my mouse is a Red Dragon 16,400 DPI laser (I run at 4000 DPI...all the time...;)) It's nothing special--got it on sale for $28. Good luck! Hope they get this sorted for the folks having problems with it!

I'VE FOUND!
I was searching to verify that I had the last Nvidia driver (I have), and saw my monitor was playing 59Hz.
I put the driver to 60Hz, and that's good!
I still think Galaxy should be improved to do the same that the exe...

Running it in XP SP3 compatibility mode solved this issue for me.

It does work in Galaxy now,
if we first at least set the Windows8 compatibility on the TQ exe or its shortcut.

It does no longer work in Galaxy with last patch, the game launches but the mouse pointer can't be moved.
It does only work when launching by the shortcut. Broken Galaxy.

Same problem is back... It's only working through tweaked shortcut (XPsp3+Admin), no longer by Galaxy.

1.43 has still this bug. When we launch through Galaxy, the TQ mouse pointer (small hand) still doesn't move.
It was corrected in a patch (1.40??) , then the bug was reintroduced.
Or it may be a Galaxy bug.

EDIT: That's good now, when we have set the compatibility to Windows 8.

This is a problem I've never noticed in the game--or the Anniversary Edition. With no compatibility options set at all except for DPI scaling set to "Application," in the last several versions of Win10x64 (currently preview build 16275), AMD RX-480 8GB, have never had a problem with the disappearing mouse (I don't recall ever seeing the problem, actually.) Glad to see you found a solution that works for your hardware.

But, I don't use Galaxy, I use the standard download version of the game. I've had trouble lately with some Galaxy-installed versions of a couple of games--but not with the standard download versions of the same games. I think that Galaxy is misconfiguring some games. Currently on version 1.45 of the game.