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.
You have not stated any reason that would appear to prevent the obvious solution (if you consider it a problem at all) of running the network wiring along the face of the wall at the same level as the server rack, or roughly 12" below the power conduit. If the basement/crawlspace floods enough to flood the server rack you probably should rethink the location of the server rack, and in any case the cables being run horizontally higher up won't help a bit in that case.
You could also move the cables far enough forward into the room (3-4 feet, say) that they don't encroach on your shelving unit, then run them back along the joist (it's holding up a floor above if it's a crawlspace, so rafter is not the right term - those have a roof above them) to the server rack.
Despite a number of "trying to be perfectionist" scare stories about needing to keep network and power cables well-separated, the fact is that the differential pairs in Cat5 (or 6, or 7) are designed to reject interference, and that the frequency domains of networking signals and power line noise are utterly different. So, in practical terms you can wire tie (just not too tight - sharp bends are bad) the network wires to the power conduit and not have any interference problems in a typical home application.
Coaxial cable is self-sheilded by design and thus also highly unlikely to pick up any objectionable interference simply by being run next to power or network wires.
Since all you are really considering is running them parallel at a few inches distance, you will be fine doing that (do avoid a sharp bend anywhere, particularly where you turn downwards and might be tempted to yank the wire tight around the J-hook.)
While I do generally try to maintain the oft-suggested 12" separation for parallel runs when designing from the ground up, I have hundreds of feet of Cat5e that is in close proximity to power wiring in old building retrofits, and connected to switches that report error rates - and those error rates are pretty much always zero, unless there is a more fundamental problem with the cable (like rats chewing on it, a sharp kink/bend, or a bad connection at the end of it.)
Best Answer
Having done a LOT of this over the years, your options are basically - find something else to follow, go plenum-rated and use ductwork if you have it as @Comintern suggests in a comment, run wires on the surface or go though a lot of agony trying not to open up walls. In many cases, opening up walls would be a lot less agony. There are often inconvenient things like firestops in the middle of walls, and you can get into all sorts of excitement you'd just as soon not when trying to drill blind or around corners or any sort of fancy work like that. Yes, long flexible drills are made - without a long flexible camera to be sure what the heck one is drilling into, you can get into a world of misery trying to use one...
My prime suggestion - open the walls, run conduit, never have to do this again (if you run the conduit right so that you can pull these wires out when you need to install whatever everything is using in 10 years.) In large part you may be able to minimize opening first floor walls if you can distribute across the basement and have short runs up to the first floor. The run from basement to attic will probably require it - you can distribute much of the second floor through the attic to minimize opening walls up there.
If you can't use plenum rated cables in air ducts (and DON'T use non-plenum rated cables in air ducts, since you and your family are the ones that will be subject to smoke inhalation for the savings of a few pennies) look around the plumbing for possible routes you might get a network wire in. However, most non-owner-built homes have miserable access to the plumbing, so that may not save you much if any wall opening. Builders who are not going to live there seem to love just sealing it all up in drywall to make more work when service is required...
Look around for things like stacked closets to try and locate space where you can open the walls with minimum area to repair/repaint to run your conduit or wires.