I recently found out that HDMI cables (v1.4 and over) have ethernet capabilities for higher res output
Yes, the HDMI protocol allows using HDMI as a transport for Ethernet. No, this isn’t dependent on the spec of the cable you use, and no, it doesn’t have anything to do with getting higher res video.
Say I have a PC/laptop connected to the internet, and a monitor (not a smart TV) connected to the PC/laptop via an HDMI cable (v1.4 and over). Does the monitor now pose a threat to my local network?
If that monitor contained an HDMI-Ethernet functionality, and your video card supported HDMI-Ethernet, and your PC enabled that interface by default, and your PC was stupid enough to route traffic between that interface and your LAN without any prompting, you would have a problem. To the best of my knowledge, none of these are true — there are no PC monitors and no PC video cards which support the protocol, no OS support for it, and in any case no routing between different networks unless specifically configured.
In the same line of thinking, if I somehow configure the same PC/Laptop (that is connected to the internet) to HDMI input,
There is no “somehow”, you can’t convince a computer’s HDMI out to become an HDMI in. It would need a dedicated piece of hardware. And while HDMI capture devices do exist… they don’t support Ethernet!