Since the cards are all based on usb-c/thunderbold, would it be possible to use them in other devices as “normal adapters”? For example, if I have another laptop without HDMI output, I could plug my HDMI card into the thunderbolt port and use it there? Or possibly in smartphones (e.g. the SD or memory card) Or are there software or hardware restrictions that prevent this?
Yep! You can use them with other devices. As long as the other device supports the same capabilities needed for the expansion card.
Some cards may have limited compatibility in other devices, for example, HDMI and Displayport both use type-c DP alt mode, which is not supported on the majority of smartphones for example. But should work on the majority of laptops supporting usb-c.
Does that mean that the Storage Expansion cards can just be plugged into the USB-C port of another computer or even a smartphone to copy or transfer files?
Yes you can plug the storage card into your phone or another device.