You cant WiFi to the screens unless they are 'smart' or have another device connected to them that supports the functionality. Depends also if you mean casting to the screens or just providing Internet access through WiFi from where they can stream TV and media. Either of them will require a device connecting to the screens to enable the functionality.
In terms of budget you can get a half decent desktop for about 250 leaving you 350 for screens, a couple of Raspberry Pis and the keyboard/mouse. Very doable.
WiFi signals and throughput will differ a lot depending on the neighbour's traffic, the hardware used and where it is placed along with the fabric/size of the home. Each channel has a finite bandwidth or throughput. If all your neighbous used the same WiFi channel then regardless of your setup you could have a very slow connection. This can be further restricted by the quality of the backbone on the WiFi router, generally the more you pay the better you get but not always.