I have a cable box and router both located in an A/V closet and a single Cat5 line running from the closet to a TV in a separate location in the house. The Cat5 line is currently dedicated to HDMI-over-Cat5 use for the purpose of running the signal from the cable box to the TV. I would love to be able to have a wired connection running from the Router the TV as well, for purposes of streaming smart-tv applications with speed and reliability. Is it possible to split the Cat5 line at both ends so that both signals (i.e., from the (1) cable box (via HDMI extender) and (2) the Router) can be run simultaneously?
Maybe. It is possible to do a 10mb connection just using 2 pairs (of the 4 available) - but it isn't recommended, or even "the right way".
If 10mb is OK for you, AND the HDMI-over-CAT5 doesn't need more than 4 pairs, then you should be good to go.
If the HDMI-over-CAT5 needs all 4 pairs, or 3 of 'em, then you need to run a second line. Hopefully the existing line isn't stapled down, etc. and can be used as a pull-through so you don't need to get up in the attic, etc.