Don't bother playing PC games with pixel shader requirements like Far Cry, Doom 3, Deus Ex Invisible War and Thief Deadly Shadows without a dedicated GPU (AKA many laptops) because they either won't look right with missing graphical elements (as is the case with Far Cry) or they simply won't start at all (i.e. Deus Ex Invisible War and Thief Deadly Shadows, based on what I've read anyway).

I play both Far Cry 1 and 2 on Intel HD4600 without any problems. In fact, FC looks much better than it looked like on my old PC with nVidia GPUs (7600, 8600 and 9600). No glitches and missing textures which I had back then.

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I have played and tried playing Far Cry many times and can tell you that some versions of the Nvidia drivers broke the game for many months. Just try with another driver version. Of course, that may break other games then...

True. If graphics are really advanced and tricky (as they whhere with thhis title), graphics programmers actually reading the white paper of different GPUs to taiIor it for different or common configurations. I ended up putting in one of my older nvidia cards into the desktop (partly just for fun). On a 560 ti and Win10 I had a smooth run, all effects seemed right as far as I could recall the graphics from 18 years ago. What made me conspicuous nevertheless was a weird offset on some more layer textures.

Why 32bit? I suppose this might be the problem.
This is just plain moronic "warning". I mean, on all of them?! Even those with GTX3080 GPUs and i7 CPUs?
If it didn't have one, you wouldn't be able to see anything, except for a black screen. :>

Maybe some very old integrated GPUs are not able to run Far Cry properly, but at least from Intel HD4xxx onwards, there shouldn't be a problem with it.
Moreover, updating graphics drivers might help. For example Sacred was a bloody mess of artifacts and glitches and updating drivers made this problems go away completely.

Sorry for the delay. As for your questions,

1: 32 bit because the physical release of Far Cry was optimized specifically for 32 bit machines. My 32 bit architecture is not the problem here.

2: I don't know, man. I don't work at Valve. I didn't create the Steam pages for Deus Ex Invisible War and Thief Deadly Shadows. But I ASSURE you that if you peruse either of those pages you will see the bulletin that boldly says: this game will NOT run on laptops! The NTSC boxes for these two games also have similar warnings on the back under the system requirements section (at least the box for Thief Deadly Shadows does).

3: What I meant is that my Windows 7 laptop doesn't have a dedicated GPU. It has graphics acceleration as a pack-in feature, but no actual GPU card/chip. The CPU handles everything, as is the case for most low-end laptops, hence why they can't render fancy-shmancy pixel shaders. That's my understanding. I don't think graphics drivers would fix the visuals if the laptop doesn't have a dedicated GPU to begin with.