This is video driver settings and display settings.
Set everywhere, including games settings where there is such option, GOG settings 9and dosbox conf fiels):
Keep Aspect ratio or streatch-with-aspect-ratio or something similar.

For display or most lkely for TV it can be a bit differ: sticher between normal\widescreen. Monitors mlabeled as 'gaming' usually have exact proportions to choose and option to keep ratio.

GOG version is DOS-only rip bundled with DosBox. There is no winmain.exe\hospita_w.exe for native Windows. In any way it just add GOGDOSBOXlauncher exe and dosbox_conf files where also should be set proepr 4:3 and check to keep ratio.

GOG releases still should be configured.
something like this
http://disk.yandex.com/d/vgEu8rmtMnK3Hg
PCGW sometimes not useful or hve wrong info, and this game is example (wrong info about Windows version issues). Still here is PCGW article
https://www.pcgamingwiki.com/wiki/Theme_Hospital
and for DosBox
http://www.pcgamingwiki.com/wiki/DOSBox

here the DosBox updated build
http://www.dosbox.com/download.php?main=1
current version a little bit more updated than GOG, you can isntall it to user AppData and at same time copy-paste dosbox files to GOG Theme Hospital folder
https://sourceforge.net/projects/dosbox/files/dosbox/0.74-3/DOSBox0.74-3-win32-installer.exe/download

Well, so I understood about a third of what you wrote about but I'll try giving it a go. Just a question - where to start? Where do I go to setup this config? Thanks.

Set lower resolution on Desktop and Keep Aspect should appear for AMD, Nvidia, Intel or for Intel+Nvidia Optimus - but in Intel anyway.
GOG's .conf file (its how it named for DosBox) should be pre-set to keep ratio. But maybe, not sure right now, its by default set to windowed. Press Windows standard hotkeys Alt+Enter to swtich to fullscreen. Screen should be partially resized to fit lower side of display with black bars on another sides to keep it 4:3.

Check conf file and read every options comments. Most important is "output", resolution, present mode = fullscreen or windowed, and aspect ratio.
And cycles. Non DosBox non GOG properly set default value and they set too much higher value as an ingame hotkeys switcher for cycles (shift+f10\f11 or something like that). Keep auto as it is, but lower "cycles" cycles.
Sounds machine settings is optional.
(i mean, cycles is related to CPU speed and affect game speed, some games cannot be played correctly even on cycles=auto, and there is hotkeys switchers for it in DOsBox, but with too higher steps, if game have differ gamemodes it requrie differ cycles - example is Dune 1 or X-com 1-3, if you see game are too fast - lower cycles by shift+f10)

During game move mouse cursor to upper side and there you find ingame game settings. One of them is 320x240->640x480 resolution switcher.

Gameplay suggestion:
set any else settings - they are reset each new game - sound volumes etc. Then set game speed to lower. And then press File-Restart mission. Now mission start on lower speed, so you have more time to build first cabinets.
Press Open clinic sooner or maybe almost at start of the mission. As soon as you hire receptionist and gp doctor - first patient will be generated at the edges of the map. But that took almost month in game time for them to come. 1 month is important, especially first ones - you pay for each month for staff and service.

Thank you, I'll probably ask my partner to help (I'm tech savvy but I feel a bit out of breath here TBH...)

Btw, I know all the tips you mentioned - last time I finally played through the game without using cheats (HAHAHA) because I used the methods you just mentioned - those a real game changer. That and building enough of GP offices and having only Professors (once they're available) in the GP offices as they are most efficient to run them (and efficiency of GP offices is the key for smooth way towards winning).

http://web.archive.org/web/20210815204141/http://images.pcgamingwiki.com/a/a0/GOG.com_DOSBox_Configurator_v2.png
sry for my b.eng and misprints. I hope ypu get what i say here. look at your default conf, my conf file. and reading comments try to edit your ones (not just copy mine as is).
As you say, you already know few of that steps for game tweaking. conf file edits is last one and same simple. more text here, less real actions actually.
GOGDOSBOXLaucnher have probably all options needed to change, except cycles tweaks. But GOG used differ lacunhers in each release, forgot tu put needed options, fix that - and forgot again. Warcraft 1 is example when even after 'hotfix' ('default options was changed and missing settings re-added to layncher') - they still forgot to make it similar to others launchers and use outdated one.
Should be output, fulslcreen\windowed, resolution - just in goglauncher. Start game from it, if desktop link lead to another simles exe file.

PCGW have only one fix mentioned and it is wrong one not needed (just unplug smartphones or wireless headsets, use USB one or PC-specific audiojack ones).

GOG version is RIP, no Windows version. That mean it can be launched only from DOS emulators. If you have normal CD version, you can try better Win-executable version. If there is issues to launch it - try WineD3D or any else DirectX (ddraw\d3d) wrappers. They will work only for Windows executable missing in most GOG's DOS games releases.

Original and most re-editions of game on CD use x16 installer that is require additional tweakig to use on x64 OS. otvdm or x16-x32 installshield\innosetup updated libraries. CD content should be copied to D:\CD\HOSP\ or something like that - and tweaked files copied above.
Or just copy HOSP and exe\dll files from cd to GOG version without overriting GOG files - add missing ones.4

There is fan-made open-source remkae CorsixTH. As any remakes - be careful with it. it have changed a lot of mechanics, although it almsot impelment everything now - something work brokenly (earthquakes is too much often cause of new random-generator feature missing that earthquakes rate is pre-set on map itself and multiply it as result - last missions in campiagn on CorsixTH is unplayable due to earthquakes)/
For old CD version you should also install official Theme Hospital patch.

Ok, I let my partner to deal with it so I'm not exactly sure what he's done (I let him read all your suggestions) - and it works. All I know he got a new version of DOSBox and changed some scripts there. Now the Hospital runs full screen - it looks crazy on a 2K screen but it's cool =)

I just forget that default start link from desktop lead to wrong file for GOG DoxBox releases. They have launchers for it. And there is screenshot of that launcher on net. I just should start with it

i put that screen here again.
If you plan in future play other GOG's DosBox releases (eg Theme Park game) - there is link file for this launcher.

There is also cyles file in .conf, and here else important settings: graphic mode, keep aspect ratio (make it ON for every settings starting from video driver), resolution.

Hey Hey...just >>> https://github.com/CorsixTH/CorsixTH/releases/tag/v0.66 <<<all problems solved.