You will definitely want to get an aftermarket cooler for that processor, Hyper 212 Plus or Evo whichever is cheaper, Xigmatek Gaia is also an option. Given what you are spending adding an SSD wouldn't be out of the question either, Samsung 830 series would be something to look into.
It looks fine, you might want to look into getting an SSD and aftermarket heatsink. You only need an aftermarket heatsink if you plan on overclocking or want the lowest temps/noise possible. If you don't want to oc then the stock one that comes with the cpu is more than enough.
I will definitely get an SSD. I don't plan on overclocking so I probably won't get a heatsink at this time.
One problem I have is that due to the layout of my apartment my computer will have to be wireless. Should I get a USB device for this or are there other internal options (inside the case)?
If you absolutely HAVE to have wireless, you need to get a dedicated wireless card for inside the computer.
PCs are not meant to run games wirelessly. Its AWFUL. I spent a year or so playing on wireless and hard wiring a cable through the walls from my modem was the best decision I ever made.