You will not have any issues using a monitor as an input device, it's a completely dumb device, so long as it is being sent the appropriate signal it doesn't care, in fact it can't care, the only thing it responds to is a basic input signal. It doesn't care that it isn't a computer, although all modern PVRs and Set Top Boxes are PVRs running over the top of a basic OS such as Android.
You will need a set top box and you will also need some kind of sound card to get sound of any description from the set top box, whether that is into a computer, or whether that is another breakout box, or an AVS Receiver, whichever way you choose to go you will be able to get sound out eventually but you will need another box as the monitor will not process any sound inputs what so ever.
You're over complicating the matter. All modern monitors do scaling, so 1080p will be a non-issue, you might even be able to get 4K output with a monitor that is capable of reaching the necessary resolution.
You need to stop over complicating the matter... plug in the set top box, set it to 1080, or whatever, away you go. Problem solved. This isn't the old days where you would plug the wrong input device into a CRT and blow it up from being out of sync, in fact most modern monitors have a much wider tolerance with regard to refresh rates than TVs and a far superior response rate.
TVs on the other hand, are not scalable, respond to a basic level of inputs such as 480, 720, 1080 and perhaps 4k while monitors will respond to every resolution in between that which makes a TV quite useless if you want to display a resolution somewhere between 1080 and 4K as most modern computers run in terms of screen resolution. The problem occurs vice versa when you want to use a TV as a monitor not the other way round.
Alternatively just grab a TV tuner card for your computer and save all the hassle and just display it on your desktop through an app.