Utilizing Cat5e and OnQ

First of all for wireless signal problem to choose a good wireless channel that others aren't using around you then use the free software iStumbler Beta.

To get cheap CAT 6 cable get it at MonoPrice.com.

For Gigabyte switch I like the D-Link 8-Port Gigabit Switch for the home.

You are going to need a few things. For starters a good router. ( I use netgear ) and probably a switch. (or 2). How many jacks in the house are you supplying with ethernet?
Your router does a few things, it will supply nat address translation, and you can also use it to supply DHCP (an address ) to each device on your network, and most routers these days have a built in firewall. (always a good thing)
And, this is just my suggestion, since your modem looks like it is already gigabit, I would stay with that on your other gear. (modem, switch).
Here is an example of a router ( I don't know how good this one is but the specs look good ) goto amazon and search for WNDR3700
and then for a switch maybe a gs108 or gs116 ( I personally have both and like them because they are fanless )

WHEW..... and as the the jack in the kitchen, you can have your phone and an ethernet connection both there but to do it you will be limited to 10/100 on the ethernet. (I have mine setup this way )

Ok, we have to determine first a couple of things. Is the module in the OnQ panel for ethernet or telephone? ( I am guessing tele) If you can take a picture I sent you my email in a private message. (well I tried to but your account must be set for not allowing PM's)
This looks correct
Incoming cable coax => cable modem => router => switch => OnQ panel

Good news if you are only servicing 3 or 4 jacks you can eliminate the switch and come directly off of the router, (that router that I showed you already has 4 jacks) You will only need a switch if you A) need more connections than 4 (depending on the router) or if at one of your endpoint connections has multiple devices, Say you have a smart tv and some other internet capable box at the same location.
Are there any unused cables coming into the OnQ panel?

Ok, definitely a telecom module. I hate when installers are CHEAP. They could have (and should have ) pulled 2 wires to each location. Anyway
All is not lost however, here is what you need to ask first
1) I just want to confirm that those cables are marked cat5e
2) It looks like you have 7 phone jacks (probably one in each bedroom, kitchen, living room. Do you need all these locations for phones? (my phone is wireless so I just use one base station with 2 extra handsets )
3) Are these jacks located close enough for your needs.
4) Does your internet connection ( coax cable?? ) come into this box?

So if you are (y, n, y, y) then you just need to get a few items
1)Router, 2) Search amazon for " Leviton 47605-C5B Category 5 Voice and Data Module 6-Port, Bracket " this may fit your cabinet. 3) when you search amazon for the previous item you will see below in the other items cat 5e quick port connector, didn't look like the best price there (they are expensive any where you get them, I either order them online or if I just need a few you can pick them up at home depot or lowes 4) Face plates for these quick port connectors. (also HD or Lowes ) 5) (looks like this is included with data module but if you use somethings else ) cheap 110 punchdown tool (less than $3)

You can also do this other ways but I was guessing you don't own a rj-45 crimper so the data port makes life a little easier. Also one more note while I was looking at amazon in the feedback section for that data module someone mentioned that it uses a cat5 "A" configuration make sure when you wire the wall sockets that you also use the "A" configuration (its marked on the side of the quick ports)

1) Yes, cables and face plates in each room are marked Cat5e


Here is a link to the Manual
http://www.legrand.us/~/media/Images/Support/On-Q/Technical-Documents/IS-000-IS-200/IS-0011.ashx
Ok, had to do some more research, this OnQ terminology is different. The WAN port was throwing me off. So from what I can glean on the internet is that this system gets setup as a phone system and then with the WAN (wide area network) you can use one outlet as a data port. Apparently the faceplates are rj45 they are probably wired straight through so maybe you want to remove one of the faceplates and take a picture and post it here. I suspect you will just want to change them and get the data module. As for the "phone jack" in the kitchen that is probably an rj45 as well I would definitely mark that (and any others you keep as phones) with a PHONE label or change them to rj11 jacks.
Also in your current setup I don't believe that red jumper cable is doing anything, You should try removing it and make sure everything is still working.

.. and just one more thing (for you and anybody else reading this)

These faceplates and jacks in the rooms that you are going to convert if they are already rj45 jacks even if they are wired straight through you could make them work. (Saving you from having to buy new ones)
Since we know the data module I gave you the link to is an "A" configuration you just have to figure out how to make the sockets match. If there are no markings inside to tell you which wire goes where in the socket( I have ran into this before) Even if they are marked you should still make sure the order is correct.
Use an electrical tester and looking at the socket ( the side the cable goes in ) with the spot the clip on the cable goes into pointing down, just wire it from left to right. Touch your continuity tester to pin 1 ( most left pin ) and then find on the back of the socket which one that is and place the Green-White wire there then #2 Green #3 Orange-White #4 Blue #5 Blue-White #6 Orange #7 Brown-White #8 Brown.