What tool do I need to cut groove along edge of floorboard
There are two possibilities - create a channel in the trim or create a channel in the floor.

For network and TV cables, you probably need at least a gap about 1/2 inch wide and 1/4 to 3/8 deep. The trim looks too shallow to put in that channel. You could cut a channel like that and cover it with the existing or comparable trim.

While a router can cut channels, it needs several inches around its cutting bits for the base of the unit, and your cut would be too close to the raised hearth.

A good alternative might be a multitool



These are specialized power saws that can cut in very close quarters. The saw blade (it has several others) is offset, so you can cut downward into the floor right up to the edge of the hearth. They come in various price ranges starting at about $60.

I would remove the molding, draw a line about 5/8 out from the hearth all around. Then I would cut downward through the floating floor using the half circle wood blade following along the line. You probably will need to use the straight saw blade at the corners.

The wood strip you cut off should be easlily removed if it is not glued down. If it is, a small prybar, cheap screwdriver or chisel should easily pop it out.

Lay your cable, replace the trim. You may want to use cable clips to keep the wire in the channel until the trim is in place.