Although it is possible to set up a local network using coaxial cables - 10BASE2 is the most popular example - is is very far from common. It was predominant in the late 80's, but has been since then deprecated in favor of 10BASE-T, 100BASE-TX and 1000BASE-T (Ethernet, Fast Ethernet and Gigabit Ethernet, respectively). These are the common network connectors you find on pretty much every computer, laptop, modem, router and network device on the market.
Apart from not being used anymore (and therefore being really hard to even find hardware available to buy), coaxial networking, specifically 10BASE2, is much slower (maximum speed of 10 mbps), have a limit of 30 computers on a same network segment, and are connected to each other using a long, unique network cable in a way called "bus topology". If a single point of failure appears, the entire network goes down.
(Image taken from The Network Encyclopedia)
As for having cable TV and PC networking on the same cable, that's impractical because the endpoints on both TV and PCs would expect the signal to follow specific protocols and to be "pure", otherwise they would interfere and scramble the communications. Because coaxial cabling for networks isn't a practical standard, I'd doubt there are relativaly modern devices designed for splitting/unsplitting the signals.