This is about videodrivers and OS.
As for launched and closed immediatly (without even window) with no errors - this is comon isue for any Windows. Most likely for OP - this is cause of Windows DEP security service. Turn it to all and add Galaxy xe as exception (it cannot be added exceptions without turn on for all apps, default system-only not allow this and yet apllied to all apps anyway). Games started from galaxy will get same attributes (disable DEP, compatibilities runasadmin, core affinity and something else)''
Anoher one - Install DirectX June 2010 +dxwebsetup - go to Control panle Windows features on\off - Legacy DirectPlay enable. And restart PC exactly as Alt+F4 - Restart, not a shutdown.
Optionally after restart chekc for latest stable updates for OS\omponents.
Start game or galaxy exe as admin. Game started from galaxy will also get runasadmin attribute.
Heroes 3 require adinrun at least for first start. ofc, if it is same as galaxy or anything else gaming related installed to system folders (Program files) - it always require adminrun, and issues still possible.
Reinstall GOG Galaxy to D:\Games\GOG\Galaxy\
and games to X:\Games\game-name\.
You can isntall game by standard GOG version isntaller (GOG Galaxy game version and client are not required) and add it after to Galaxy client as GOG Galaxy build.
If you still prefer all do in client - next to game dowanloding you should do game verification. At least twice. Or better third times in a row - GOG Galaxy currently no more report result of verify and always redownalod something (appconfig related cfg-registry things, OS redists).
Anything of that can help. First thing - is just 'Restart' OS.
Last one is Heroes(x)GL wrapper mod.it not solve launching issues - it is QoL improvments. Like Heroes3HD+ but IMHO - better. Less modification. GL wrapper-mods from Verok require game to be launched and configured first.
Also, if you plan to se it - dont forget to remove compatibiility applied by GOG. Better way is copy and rename game exe (and set nw exe in Galaxy Library in game cnfigurating).
More solutions here and on PCGW articles:
http://support.gog.com/hc/en-us/articles/214062465-Heroes-of-Might-and-Magic-3-Complete?