OK, this is kind of a rambling question with a lot of details, but let me give you some leads on some of them:
Wall Plate
If you want to install a wall plate, you'll want to clean up the opening and then install a "low voltage mounting bracket", which is a square piece of plastic that frames the opening and provides a place to attach the plate:
(Wall plates come in sizes like 1-gang, 2-gang, etc., which indicates how many toggle switches or receptacles it can fit. You might want a 2-gang for that size opening and number of cables).
Then you can get a "keystone plate", which is a wall plate with small square holes in it (available in various widths and with different number of keystone holes). Here's a 1-gang, 2-keystone plate:
Finally, you get different keystone modules for the different types of cables. They snap into the holes in the plate. This lets you mix and match different connection types on the same plate. I'm not really sure what you mean by "tv type" and "telephone type", but here's a coax keystone module:
(EDIT: resist the temptation to skip the bracket and screw the plate directly to the wall. The screws will pull out of the drywall in about 2 seconds. I've seen this happen in several houses where people added coax wall plates or something, and didn't bother to do it right.)
Shelf & Mounting
As for wall-mounting options, it depends on how tidy you want to be. You could either mount a small shelf to the wall and place the electronics on that, or use your pegboard and find a way to strap the items to it.
Personally I would mount the shelf or pegboard just below the wiring plate, not on top of it. That would make it easier to mount the wall plate, and easier to make changes in the future.
Whatever you mount to the wall, make sure you attach it to the studs behind, not just screwing into the drywall.
Best Answer
It is a Ceramic Insulator for heat protection - the wires go to the heating element.
EDIT 5-8-2017
As per Ed Beals comment (Thanks for the FYI) - I will make some explanation here .
Heating elements are exactly that they heat up as the electricity flows through them, they have different ratings of wattage consequently also heat output, How hot they get. Because of this there needs to be a method of protecting the wiring of the circuit from the transference of this heat.
The heat transferring via the wire into the insulation can melt or make brittle the insulation and thereby create a potential fire hazard and or electrocution hazard. Ceramic Insulators are used to dissipate the heat and prevent this transference that would cause the insulation of the wires to melt or become brittle.
Over time the heat will cause these ceramic insulators to break down - constant heating and cooling causing them to crack and become brittle and break. You need to use them in the replacement of your part and in connecting the wires to the heating element - unless you would like to risk the lives of the occupants of the home to fire.
I should also add that sometimes the heating element itself has this component - and therefore a new element would need to be purchased.