Mine always happens without sending people to research - at some random point in the game, the door disappears. A patient is trying to walk in, just just walks endlessly on the spot next to the door, even though the doctor is inside waiting for them.

So frustrating as it just seems to happen with no pre warning and have to load a save game, only for it to inevitably happen again on my next game. Literally making it unplayable.

How to ask GOG to fix this problem with that patch?

This did NOT work for me.
I wouldn't keep my hopes up, but there are some workarounds as already found in this thread:

1. Save often. Make it a habit (instead of an obsession...) by usually saving at the start of every month. Since certain things happen, like monthly bills/salaries and new staff available, those are a good reminder to save. But you can simply keep "Autosave" enabled in Options and your #8 save slot will be automatically updated often enough.

2. Don't put plants in any GP's Office (with backup saves you don't need to do this, but it's safer).

3. When a [?] pops up, never immediately click it. First save, then locate the patient and wait for them to leave the room, which should allow you to pick any of the options and without reloading.

There is autosaves (per month?), Ofc manual saving always mandatory in any game. Auto initially and after consoles now mean saves for "Continue" feature.
You get busted in NFSMW → auto saved.

"Mission # start" as seapare save. And time to time to at least two differ saves. How often is up to you. Switch between saves. Whatever game and how it can be bugged. This is how it indented to be in the first place and why we have 8 save slots.

As for avoiding bug. 3. is most important I guess. Same as picking patient and press on his panel "stop diagnostic and try to treatment". Wait until they completely get out of any cabinets. For me it happens on [?] fax screen too, when I choose " try to cure anyway". Probably I open that screen too fast.
I play on medium speed (with slowest on mission start).

BTW, fax screen did not pause the game. Still not only for decide, but for fax screen itself wait until patient get out.

Make sure (I didn't) game do play on FPS not higher than 29/59/119. Experiment with that. For GOG/EA App version with DosBox it can be issue with CPU cycles. Keep it auto, but edit gradation and correct it during gameplay by Shift-F11/F12 (change cycles by gradation set in config). Still that bug happen in winmain.exe version too. Limit FPS for the game to eg 59 by video driver control panel or probably better-easier - by RivaTuner RTSS tool.

There is Sony PlayStation PAL MULTI version (usual M5 + Svenska as well as in PC version, + Jap version separate). As alternative. Make sure emulator correctly apply PAL fps limit.
For Chinese and polish - only PC version.

Or CorsixTH, but it mostly for skirmish/[s]multiplayer[/s]. Mod-remake still not bring all campaign features and probably still suffer from "double" earthquakes issue.