I regulary have the issue that RT3 will start, but not create the game Window. It runs in background, plays music, is in the Taskmanager. But there is no Game Window to switch too. The cursor does switch to middle of screen during start, but no game window materialises. The start appears also a big faster.

It does not seem to follow any order - just start and kill via task manager will eventually give me a running game.
So I asume it is a Race Condition of some sort. Especially as the game uses Threading heavily (for such a old game).

If it runs properly, Railroad Tycoon 3 on the main menu has the following load (according to task manager):
CPU load/Memory/Threads
25%/35540k/10-12 Threads
25%/35336 k/8 Threads after the music stops

Bug case:
25%/32.336k/8 Threads
25%/32.296k/6 Threads after music ends

All I can tell is that in the error case, 2-4 of the threads appear to never start. Hope that helps to nail down the cause of error a bit.
The CPU load just equates to one of my cores.

My Computer:
Windows 8.1, 8 GiB RAM, Intel I5-4200, HDD
GPU(s): Intel HD 4600 (from the CPU), NVIDIA GeForce GTX765M. The game naturally runs on the Intel HD 4600 as it is by far enough.

I can't get this game to run in Win7 either. When I start up the game, it changes the screen to a smaller resolution, but there is no game window. Is there any way to fix this?

Are you guys running the game as administrator? I got similar/same error, in Win 7, when I didn't do that.