How can one properly and tidily install this hardware for internet access?
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 &amp; 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.