Could you provide more details about the internet connection (specifications)? Maybe I won't have to buy anything... (is it really wireless - WiMAX?)

The advertised speed is up to 15 Mbps. But the actual speed I observe at my home is 2.5 Mbps download and 1 Mbps upload. For comparison, a regular 3G connection provides 1-1.2 Mbps download and 300-400 kbps upload. It's enough for me, but I'm not sure what will happen if I connect 20 computers to it. There are no problems with playing games, but I'm not sure about streaming. At least, disconnecting a third or half of the load from the main source would be beneficial.
And a Wi-Fi modem.

I didn't quite understand what you're planning to buy and why?...