I've been gradually introducing RPis into my teaching over the past couple of years, and it now features heavily in my N5 and AH lessons.
Initially I was unable to connect these RPis to the school network, but eventually I persuaded our IT dept to modify an Ethernet port in my classroom to isolate it from access to any other machines on our internal network. This port now gives access to the internet only, going through our firewall (and therefore provides all the standard filtering -- blacklisted sites/categories, etc.). I have physical control over that port, and plug into it only as needed--a key argument which helped to convince IT to set this up for us.
Connected to that one port I now have a cheap router, which is connected via ethernet cables to a set of 8-port unmanaged switches (£16 each), which in turn provide an ethernet cable to each workstation seat in two adjacent computer rooms (total of 45 seats). This gives us a second LAN within the CS dept., which can be connected to the internet if I wish it to be.
In lessons where RPis are to be used, pupils take a Pi to their seat, and connect it to the CS LAN. We use the existing USB keyboard/mouse from the Windows PC at their seat, and the PC monitors there each have a separate HDMI cable plugged into them, which can be inserted into the HDMI socket on the Pis. Power to the RPi is supplied via a USB / microUSB cable plugged into the side of the PC monitor by the pupil.
This environment works very well, and has proved especially beneficial for AH pupils. For N5, the RPis are primarily used for the Web component of the course; we are currently using Visual Basic for N5 programming, and MS Access for DB, so cannot use the RPis for those. In AH, a great deal of their work can be done on RPi (Python, PHP, web server, etc.) so it is their primary environment in that course.
One side benefit of using RPis in my classroom is that I can install/configure the software on Raspbian as I decide; on our Wintel PCs, that is totally under the control of IT, whose priorities are not always aligned with our needs. The downside of that, of course, is that I have to do all of that installing/configuring, but it's still a net win.
The pupils really enjoy using the RPis, and I enjoy teaching CS with them.