Is it possible to use my laptop as a monitor for my new PC?

Unfortunately, you can't use a laptop as a monitor. Laptop display ports are outputs, not inputs. The good news is that monitors are pretty affordable these days, so it might be worth just getting one.

You could technically take apart your laptop and connect a display port directly, but that's quite complicated. Alternatively, using a capture card might let you show your PC's output on your laptop, although it could add noticeable latency.

Almost definitely not possible. Laptop screens don’t function like monitors do. Luckily, prices for monitors have dropped significantly, so finding a good one isn't as hard as it used to be.

Generally, most laptops don't support being used as monitors since they lack display inputs. Even if yours did, it would still need to be powered on. Without knowing the specific model, the chances are slim that you can make this work.