

There are kind of two different questions here.
First of all yes, humans are pretty good at telling whether someone is looking them directly in the eyes or not. So if you were to ask someone directly whether you are looking in their eyes or some other part of the face, they would probably be right most of the time.
The second part is whether they would notice it consciously without bring asked. That’s a little trickier. I suspect if you were staring directly at some other part of their face, people might get self conscious (do I have a zit? Is my nose that huge?) But looking at a spot between their eyes, or shiftng your gaze periodically would probably fly under the radar.
The only thing to watch out for using a laptop that is plugged in 24x7 is the battery. Battery management systems are generally pretty good, but Li-ion batteries can fail catastrophically. As long as you make a point to check on it periodically it’s probably fine.
I’m using an old laptop as a local interface for my network setup, since its in my basement, and I actually pulled the battery out entirely since I have a beefy UPS powering everything. Paranoid, maybe, but a Li-ion battery sitting on top of my equipment rack could do a ton of damage if it were to fail someday.