The free version of Edge Inspect limits you to connecting to one device concurrently. (You can certainly disconnect from that one and connect to another if you wish, but it's just one at a time.)
If you purchase an Edge Inspect subscription, or have a paid Creative Cloud membership, then you can connect an unlimited number of devices concurrently. (Note that I have seen up to 35 connected at once.)
For info on purchasing options, see this blog post: http://blogs.adobe.com/edgeinspect/2012/10/23/how-to-purchase-edge-inspect/
It does not have to be a decicated wifi router, although it certainly can if that makes sense for you. For example, at my home I have a single wifi router that I always use for my laptops and video streaming devices, with my desktop computer connected to the same router on Ethernet, and I can connect my 5 tablets and 2 mobile phones to Edge Inspect on the same router with no issues.
My router says it can support up to 250 connections at once (Ethernet + wifi); however, I have nowhere near that many devices so I could not put that to the test. I think a more reasonable limit is around 30; however, that is just based on having to share bandwidth. You might want to check what your router says is its limit.
Just keep in mind that the more devices you have downloading data, the slower the performance will be.
To add to what Carey said above, the wi-fi /bandwidth limitation you're most likely to run into if you connect 20+ devices is with screenshots. If you have that many devices connected and choose to take a screenshot of all of them simultaneously, they'll all be trying to transfer the screenshot data simultaneously and you'll likely notice that both the devices and your computer will seem to be unresponsive for a period of time because they're all so busy. The simple workaround is to take a screenshot of one device at a time.
Thank you, this was very helpful