justanoldgamer: Syndicate Plus is supported in Windows 10, I only checked that one but I suppose the other DOSBOX dependent games are also supported.
As WVPR says, check the dosbox version. I would add to check if you have the latest GOG version of the game, I think Galaxy is suppose to take care of that (I don't use it) but it does not hurt to check. I know you said you also downloaded them directly but you don't say when. (I hope I do not sound rude but 3 decades of professional and amateur tech support thought me to not assume anything)
The latest official release of DOSBOX what 0.74 in 2010, I think it was in the Vista era. GOG says the game is supported on Windows 10 which I assume means they tested it. If you have the latest GOG version and it works on their Windows 10 machine then the likely culprit is your computer (including stuff like video card/chip, sound card/chip, and drivers) and the software it runs in the background like but not limited to anti-malware (anti-virus, firewall etc...)
By saying it is your computer's fault I am not saying it is your fault or that GOG is not responsible, I'm just saying that computers are amazingly complex beasts and I spend my days amazed that they work at all. For example, there is a game, I forget which one right now, who won't run with the driver of my logitech mouse active, I have to use window's generic mouse driver for the game to run.
Not running Windows 10 I can't really help you but providing all information you can think about your machine might help someone who can.
So, what is the brand name if any, memory amount and speed, cpu type and speed, memory type and speed, video card model (or integrated video model), same thing for soundcard, mouse. I've never seen a monitor causing a program to cause the type of issues you're having but throw in its model just in case. Antivirus, firewall, programs that you're always running in the background like a music player or a scheduler reminding you to eat and sleep, anything but the computer's color and your favorite beverage while gaming (do share if it's a good beer).
If you do not know some of that information or how to get it, that's ok, if it is important I'm sure someone will help you figure it out.
This post written under the influence of the following beer:
First of all, I appreciate the tech-support-career-derived diplomacy in your tone. That said, I am also fully aware that this is likely an issue on my end, and am happy to get any feedback on what it might be, especially if others have encountered similar issues.
So, let's go down the list.
1) I downloaded the latest installer for Syndicate Plus directly from the website after experiencing the crash on the GOG Galaxy version. It had the same issue, where the DosBOX splash image will appear, cut to black, and then the program stops responding (this is the same issue with the other DosBOX games). Checking the DosBOX version for Syndicate Plus, it's 0.74.
2) I'm running:
Phenom II X2 555 Black @ 3.2GHz
12GB DDR3 1600 RAM (two different manufacturers, and I honestly forget which)
XFX Radeon R9 270X 2GB (I DID just update to the latest drivers for this card)
ASRock M3A770DE Mobo (I'm using the integrated sound)
Logitech G502 Mouse
Logitech G35 Headset (mentioned because it integrates with the LGS suite)
LG E2241 monitor
Programs:
Logitech Gaming Software (8.57.145)
Logitech Profile
EPSON Status Monitor for XP-200 printer
Google Chrome
Adobe Reader Updater
GOG Galaxy
AMD Catalyst Control Center
OneDrive
Google Hangouts
Regarding A/V, I'm running Windows Defender. No third party solution right now.
3) Beer-wise, I've been more of a La Fin Du Monde than a Maudite type when it comes to Unibroue, and I'm a stout and porter drinker in general, but I used to go nuts for Dogfish Head's Theobroma.
I hope some of this information helps!