I had some issues with my broadband connection in Qatar. I called the customer service and the chap at the other end kindly helped to restart the router and some other device(that also looks like a router and fixed to the wall permanently). He then said that I needed to change the wireless password by logging to the router and said he could change it for me to whatever I preferred. I declined his offer and changed it myself. There were a number of things that worried me from this experience

• The chap from the call centre knows what the name of my wifi network sitting remotely
• He can change the password of my network from his desk remotely

From your experience in the region is it common for ISPs to be able to monitor their clients, their wireless networks and change passwords? While it is bit scary to think they can do it, is the view that I should have been aware of this and my internet access should be limited to 'sensible and non-controversial' websites?

Use a VPN if you're worried about your ISP checking your internet usage.

The router likely has remote access enabled so they can do that sort of thing. I bet they get hundreds of calls a day but assuming you can log in to the router and access its management screens you can likely disable that.
The other box is likely just the cable modem to convert the fibre or phoneline to wired ethernet.