Reddit mentions of TP-LINK Power over Ethernet Adapter Kit (TL-POE200)

It's possible, but not advisable.

First off, real PoE is a complex protocol involving negotiation. It's intended to automatically sort things out when PoE and non-PoE devices are all plugged into the same PoE switch, i.e. power the things that need power and not fry the things that don't. There is also a lot of hardware out there that sends power over ethernet without actually obeying the PoE spec, for example this.

Supplying several voltages over the wire is tricky. If you draw a lot of current at one voltage, for example, the voltages on those lines will sag and the voltage of "ground" will rise, which means that 5v will become less than 5v even if very little current is being drawn on it.

You are best off running 48v through the unused pairs (two wires for each +48 and gnd) and regulating it down to whatever you need at the remote end.

I've been using this kit with my Pis, it works well.

For an office environment, I would recommend NOT going with a basic router. This will lead to issues of its own, especially in the budget you are talking about.

If you must, though, they sell a POE injector that you can place in line at the patch panel, and another in line device that goes at the device, allowing you to plug a power cord into the devices power port. Poor man's poe. https://www.amazon.com/TP-LINK-Power-Ethernet-Adapter-TL-PoE200/dp/B004UBUB7C

I would recommend looking into a slightly bigger budget, and look into something like an Aruba IAP (instant AP). Business class wifi, with a cloud based management system. No need for additional controllers, and easily scaleable.
http://www.securewirelessworks.com/Aruba-IAP-103.asp?gclid=Cj0KEQiApqTCBRC-977Hi9Ov8pkBEiQA5B_ipSOw76JDk6YHimcy8SnZY6asXFPrLJaCkxFXiqMdBD4aAnRf8P8HAQ

https://www.amazon.com/dp/B004UBUB7C/ref=cm_sw_r_cp_awdb_PXs5yb32FAXN5

This kit is what I should be looking at then right?

There are three kinds of PoE.

1. "Passive" PoE. Essentially you put the DC adapter for the unit near your switch, and the cable acts like a long DC extension cord. Potentially dangerous, not recommended, but VERY cheap and if you only need to go a few feet it's probably ok. https://www.amazon.com/iCreatin-Passive-Injector-Splitter-Connector/dp/B00NRHNPUA
   
   
2. "Regulated" PoE. This is basically 'passive PoE', but instead uses a higher voltage (typically 24v) on the injector side, and then some kind of regular on the receiving side to ensure that there is not a voltage drop for long distance runs.
   
   
3. "Active" PoE. This system, which is standardized as 802.3af (as well as others) uses communication between the injector and receiver, so that when you plug a non-PoE device in, no power is sent over the wire. This makes it much safer. This is the best option, but also the most expensive. https://amzn.com/B004UBUB7C
   
   If you only have one or two PoE devices, then individual injectors works fine. If you're going to do a big camera setup with 4+ cameras, then you're better off with a PoE switch or a 'midspan' (which is just a poe injector for multiple devices). I highly recommend you stick with 802.3af stuff (aka 'real' PoE).

I don't know of any way of converting it, but depending on your mounting constraints something like this could be adapted to make it work.