There are some internal cablemodems, but they aren't on most cable companies' lists of acceptable modems. They may be DOCSIS compliant, but that's only the first level of compatibility. The cable company has to be willing to support the modem with config files, and firmware updates. An internal cablemodem also has the disadvantages that it cannot be separately reset, and would also require OS-specific drivers. So even if the stars align, and your cable company allows any of the internal modems available, you'd be paying a high price just for cosmetic purposes.
Most OS's can also act as routers. Windows has Internet Connection Sharing, for example. With multiple NIC's, and the right software, it can be done -- just not well. The computer's primary purpose is to run applications, and while pre-emptive multitasking is a wonderful thing, it's not a great environment to be running network routing services at the same time. And if you have an undersized case, there probably isn't room for multiple NIC's.
The simple, consumer-level "broadband" router combines basic router capabilities, and provides a basic firewall as a side effect of using NAT. NAT means that each computer on your LAN has an internal IP address, and the only IP address that faces the rest of the world is the single, shared address of the router. Thus incoming packets need to be expected so they can be translated to the right internal node. A true firewall goes far beyond NAT, and would be a separate box. However, based on the questions you're asking, you are highly unlikely to be in a category that requires more firewall protection than a NAT router can provide.
There are cablemodem / router combinations, however they also have deficiencies that far outweigh any cosmetic reasons for wanting them in a single device.
If aesthetics are so important that you just can't have these boxes near your desk, Cat5 Ethernet cable can be up to 100 meters long, so you can always place your cablemodem and router in another room, and have nothing but the cable coming out of the wall.