I find 4k pointless in monitors, they're too small. Some argue that you should just enable DPI scaling, but that's a lame excuse for a pointless product.
DPI scaling not only introduces GUI bugs for programs, but it's also trying to simulate a lower resolution. Yes, you can use them with various programs that will allow you to change the size of text and icons and so on, some customization even natively.
4k gaming on the other hand is pretty sweet, but as long as text is involved, and the game wasn't designed at that resolution for small displays, you run into a problem. You might as well buy a magnifying glass when buying a 4k monitor - you're gonna need it.
Input lag in a TV isn't that bad. Even if 30 ms is noticeable to some people, using them with a mouse at a high refresh rate basically gets rid of that lag completely.
Now as far as 1080p on a 4k display is concerned. There's no short answer. As mentioned above, there are multiple ways to scale, but currently no display in the world is able to display 1080p on a 4k display without losing quality in the signal. There's no distortion however, you're not stretching the image more on the x-axis than y-axis. The aspect ratio remains the same.
Also, I wouldn't say that the primary reason TV's are expensive is because of the scaling algorithm used. I would say it's more about the panel specifications. Not all IPS are equal, not all VA are equal. Not all TV's are 60 Hz, majority of 4k TV's are currently 120 Hz natively, some even let you use them with a PC to output 1080p 120 Hz without frame skipping.
Basically, when you pay more for a TV, you know it (depending on how much you spend) has:
- More HDMI ports
- More formats supported (HDR10/Dolby Vision)
- Faster processing (smart TV)
- Higher contrast ratio (and black level) -- If VA
- Lower defect ratio (better uniformity, LED technique is full array, often local dimming as well)
- Lower input lag
- Higher refresh rate (faster response times and support for 24p playback etc.)
Just a few examples, but the same can't be said for a monitor. The second you factor in display size, you throw the argument of a monitor being better value compared to a TV out the window.
All the best!