I´ll try installing the Steam versions then, see if there´s any difference.
Will do. Since that´s DX5-7, it would only apply to C47 right?
Like @EvilGeniusRo suggested, creating a Save folder should fix that:
Great thread you made there 
I´ll probably try following it on the next reinstall 
(I mean I always go to PCGW for the troubleshooting, but this is a nice step by step guide)
A little update though on my issue. I wondered whether it could be that the games for whatever reason are not running on my Nvidia gpu but the Intel one instead (even though I specifically assigned the Nvidia one to them), and found this guide: Definitive fix for this game (Cutscenes, FPS, Widescreen, Crashing) :: Hitman: Contracts General Discussions, which mentions:
Game using my iGPU and not my dedicated GPU/Low FPS (Using D3D8)
If you have a laptop, there’s a big chance the game won’t use your dedicated GPU when using D3D8 mode (refer to previous question, RenderD3D.dll). There are 2 workarounds for this: You can improve GREATLY your performance if you go to the in game graphics options and disable Post Filter, or you can do the next.
First, download this widescreen fix http://ps2wide.net/pc.html#h3 and extract h3.7z files inside your Hitman Contracts folder (password: h3). Then, modify h3.ini and put the resolution you want. And now, you’ll want to download a d3d8 to d3d9 wrapper. Just download this: https://github.com/crosire/d3d8to9/releases (d3d8.dll file) and replace the one in the game’s folder with the one you just downloaded. Voila! Now you can have d3d8 wrapped in d3d9 functions, and the game will now use your dedicated GPU, so no broken cutscenes and no bad framerate. You can enable those post filter effects again, too!
So, having already applied the widescreen fix, I downloaded the wrapper and replaced the d3d8.dll file, and indeed, the lag and framerate drop was gone. However, since this replaced the d3d8.dll that came with the widescreen fix, the fix no longer works, so fixing one issue leaves me with another 
I´ll have to try to tinker with it again, maybe in a different order, but anyways it seems like the D3D8/9 + GPU is at the core of the issue…