Are your wall mount racks hinged on one side? If not I'd consider getting those, it makes it easier to wire up the patch panels.
I would try to route the cable grouped by type / use, and if you use velcro you can easily loosen each strap and add in new runs with the existing ones where they run along side each other. Some kind of mount to hold down the velcro straps like BraveSirRobin pointed out would be handy, though you could just slap a wide staple across the strap too. You can also go to full on datacenter-style wire management stuff with fancy box-style conduits that have flip open covers and slits to pass wires through, etc.
If you want to run the cables "out of sight" you might rig a large hinge / set of hinges to hinge the entire mounting board off the wall, so you can get to things. Obviously it'd be easier to run in front but it depends on how you want to do it. I myself would probably just run them in front. If you ran them out of sight then you could make openings above or to the side of the various things such as the cat5 patch panel to let them pass through close to where they terminate.
For the router / etc, as I said before you can bypass the Verizon supplied Actiontec router and just use your DLink, just run cat5 to the FiOS box for the upstream port, and call Verizon to have them switch the port the Internet is on.
You may wish to consider placing things such that you can easily fit in a large (12/16/24/etc) switch or possibly two - Gigabit is nice to have but more expensive, could have an 8/12 port GigE and a bigger cheaper 10/100. Just plug the 10/100 into a port on the GigE if it has a GigE upstream port or if not either the GigE switch or router directly. Position these such that you can use short (.5-2') patch cables to on the fly reassign cat5 ports that are terminated at the patch panel to individual switch ports.
This gives the option of overwiring the network connections now, once, and upgrading the switch(es) later easily, or changing what you use which port for. Put something GigE capable in later (perhaps just a small switch w/ GigE uplink for the AV rack?) and you can switch it from a 10/100 port to a precious GigE port in just seconds. Prewire extra cat5 in places you aren't using it yet, just in case you might later - then just go back down and slap a patch cable from the previous unused port on the patch panel to whichever switchport.
My friend wired his house like this, more or less, though its only something like 12 cat5 runs maybe ? But he ran extra connections in the living room etc so that if we have a LAN party we can wire into the wall there and just add switch capacity at the wiring closet end (he doesn't have a switch big enough to use all ports at once, since he doesn't need one yet, but if we LAN, any of us can bring a small switch to add in temporarily - we got 'em). Fewer spaghetti cables going everywhere, less tripping
And of course you can patch in POTS lines the same way, if you get a POTS patch panel sort of thing, so you can actually convert a port from phone to ethernet by just rearranges some wires at the patch panel.
/ramble
