What kind of monitor for my laptop?

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3. 4k monitor or 4k tv?

If it was sold as a 4k monitor, then it should have connection options more than just HDMI. Regardless, what happens next depends on how good the graphics hardware is in your computer.

Worst case is you might need to buy an adapter to physically connect them. And then the graphics card might not be able to drive the monitor at full resolution. Or high resolution works great, but is too slow for games.

Try it and find out! Or read up on it, here's one reference: https://www.techcenturion.com/intel-hd-graphics-guide