You have good answers so far, but I have several items that I don't think have been covered yet. I will start by recapping.
Number of Cables per room
Cat5e (and cat6) can be used for telephone, both the old school phone and VoIP phones. This makes it easier to not have to worry about running as many different types of cables. I would highly suggest going with 2 cat5e + coax to every room and maybe even more to "special" locations. I use a TV service that can run over my network and the extra switches in each room add enough delay to cause the TV service issues. Having 2 connections in each room would have allowed me to plug the TV receiver in directly with out the extra switch, unfortunately I only have 1 cable per room so I have had to be creative with my setup.
The special locations to add additional cables would be places that you plan on having lots of electronics, like maybe in your living room. In my living room I have the TV, XBox, Wii, Blu-Ray player, TV receiver, and media center PC all with network capability. 6 lines probably would have been over-kill, but 4 lines would have been nice at that location. Some people also like to add extra coax in rooms that they have TVs. This can allow you to use multiple technologies such as satellite, cable, or antenna.
In a prior house I added 3 cat5e and 1 coax to every plate. Every room got 1 plate except for the living room and the office which both got 2 plates. This setup worked out very well for me.
Location of the "Rack"
As for the location of the "server closet", a patch panel does make things much easier to plug in. It also makes it easier if you need to connect it to an old school POTS phone line. There are some structured wiring solutions (see examples) that will make your wiring a bit more geared toward home use over a patch panel. There really isn't and huge advantages of one over the other though.
In a prior house, I added network cables and a patch panel to a closet. This picture was taken way before I finished the job, but you will get the general idea of it. I dropped all of the cat5e out of the left conduit, the coax out of the right conduit, and the middle conduit was the "service" lines (ie cable from the phone company, cable company, and eventually my antenna). The conduit didn't really go anywhere other than up into the attic, it just provided a neat way to transition from the attic to the closet while being able to get above my insulation and not having it falling down into the closet. I eventually used some spray foam in the conduit to seal off the heat that I had leaking into the room from the attic.
I then mounted my router and wireless access point to the wall.
There was already a shelf that is just off the bottom of the image that I placed things like my battery backup on.
I would recommend thinking about where your electronics will be placed. Will they be able to get enough circulation with cooled air? The last thing you want is to be replacing a modem every few months because it keeps over heating. You will also want to make sure that it is some what centrally located. This will help keep the run lengths down and thus save you money. It will also allow for you to place the wireless access point near the source of the rest of your equipment while still being in the center of the area that you are trying to cover it with. The pictures I showed were in a front entry closet that was very centralized. I had great WiFi coverage, but it would get pretty hot since there was no circulation. I did have 1 router fail, but I am not sure if it was because of the heat or not.
Distance between Power and Data
This distance between power and cat5e isn't a huge deal. 1 foot apart is probably more than enough, but to be safe you could go 16 inches or more. I am not sure how your house can have no studs, but getting it to the other side of what ever is supporting your walls is typically plenty.
Future Proofing
If you are really wanting to be future proof, you could run conduit to every location that you have Cat5e running to. This will make it very easy to add new cabling if in lets say 20 years everyone is running fiber to everything. It will add more cost, but potentially you could run less of the "just in case lines" and wait to run those until you actually need them. Probably wont offset the cost completely, but at least something to think about.
Depending on your setup, a straight run of conduit into the attic might work perfectly fine, or you might need to run it all of the way back to the "rack". Whatever the case is, make sure you will be able to easily get new cable into the conduit. This may be leaving a pull line in the conduit or enough room to use a pull rod in the conduit.
Personal Thoughts
As a side note, my personal feel is to put everything wired as I possibly can. This makes the performance of my wireless devices (laptops, phones, etc) much better and the wired devices much much better. As for the switches used to get up to the number of hardwired connections you need, the rule of thumb you should follow is the less switches the better. You would be a lot better off spending some money on a single larger switch than to have a bunch of smaller switches wired together to get you up to the count you need. It will typically work having many switches, but it is a nightmare to troubleshoot if anything starts acting up.
Best Answer
Those POTS line look like CAT-5 Look on the cable jacket. If so you can re-terminate the ends and be fine with them. The whole thing looks too new to be the original old-style CAT-3 "voice line" or the even older 4 wire bellcore CAT-nothing wire.