Ok, you've added a piece of information here that is of interest. If the game will run on your laptop without issues, then by rights it should run on a desktop without issues as well....assuming you are using the same installers for RS2.
Which means that the problems could be caused by something else.
First off, you should set your video card AA setting to 'allow program or game to control'. RTW does not like AA for some reason. It used to work, but at some point it became a problem and started causing CTDs. So disable it completely. The textures in RS2 are some detailed that it's unnecessary anyway.
Second, ensure that the fans on both processor and video card are working, and that the heat sinks are not clogged with dust. Inoperable fans and dirt will cause overheating, and the graphics in RS2 on the battlefield will DEFINITELY cause heat. I completely toasted a video card with RS2, so I know. And I was just testing a lot, not even playing much. If processors and\or video cards start overheating, they become unstable and you'll get a crash in the game for no apparent reason. It's very easy to overlook this issue....a lot of people never even open a desktop case, or would be afraid to.
Regarding the 'Steam Only' discs....well, I agree that's dumb. What the heck. I'd be pissed. But, the first version of Steam RTW actually worked with RS2. It's the current one available from Steam that does not. Their exes were altered by Steam or CA or both, and they will not work with RS2 at all. I'm not happy about that, but they pay no attention to me.
I'm truly sorry you're having so many problems.....it really annoys me that Steam and CA have screwed around with RTW so much with all these versions and changes that have broken a number of things. I just can't even keep up with them. I have tried to buy every known version of this game and test it. But up to this point the only one I KNOW will not work is the one Steam put up lately.