The ports on that NVR are POE ethernet ports for cameras that use a wired ethernet connection, and the POE means it can send power over the ethernet cable as well as the data.
Since you are using wireless those ports are pointless though
It is the 5th port on that model that you connect to your existing network. In fact if you hooked two cables from your router or switch to that device at the same time you would cause a spanning tree endless loop and slow your network to a crawl if your router/switch does not have the feature to detect this.
As far as enough bandwidth with one connection it is not a problem at all.
Check this link:
http://stardot.com/bandwidth-and-storage-calculator
4 IP cameras with H264 encoding with 720P HD resolution at 25 frames per second (which is a pretty good camera) is only 19kbps for all 4; if you only have a 10/100 network connection then that is 100mbps, if you have gigabit it is 1000mbps, either way not a problem.
If you want a standalone box that is fine but it is not necessary. I have a desktop PC and use Blue IRIS software ($50) and it records to a hard drive and works really great, and has a good android/iphone app as well, and of course you can view a camera over a web browser as well. It is really versatile for the low cost point, you can even have multiple logins and profiles, motion and alarm triggers, scheduled recording, etc, etc; I would bet it has every capability the software on that Q-SEE has and then some.