Does amount of display's pixels affects computer's performance on lower resolution settings?
Setting your graphic card to work at a lower resolution will provide a small increase in performance in the case you described. 

For example:

1920 * 1080 = 2,073,600

1280 *  720 =   921,600

2,073,600 - 921,600 = 1,152,000, or 56% fewer pixels.


If you spend all your computer's time working to create text to fill the entire screen with new information then you would be able to do that almost 180% faster - but that is an extremely rare use case.

One way to understand how that's a rare use case is: when you look at this webpage it's the graphics card that updates your screen (and not your computer's CPU), for 1/10th of a second your computer will be faster and for the remainder of the time you will gain no speed increase (and suffer with lower resolution).

A more realistic way to estimate is that if 10% of the CPU's effort went into creating a screen full of text, each second, then you would speed up that 10% by almost 180%. Put another way, your computer would be 8% quicker if all you did was have text go flying past your eyes all day.

It's not a good tradeoff to suffer too low a resolution for a small increase in performance.

Don't confuse the above with watching videos. Watching videos involves decompressing a highly compressed data stream, most of the effort is running the algorithm to undo the compression; writing the pixels to the screen is the easy part. In fact a lot of compression algorithms rely on the fact that not every pixel changes constantly, deciding what can be skipped and what needs to be updated, along with how to reduce the data stream's size, represents most of the effort.

Going from 1920x1080 to 3840x2160 is a much bigger jump, and in fast paced video games where everything is constantly changing indeed there's a lot of effort going on in the background to create new data to display (or decide if it would not be visible, and not display it).

Yes, you'll get a bigger jump in performance going from 4K to 1080p in a gaming scenario but at a huge loss in resolution (and enjoyment); it would really be a waste of money to buy a 4K capable computer and run it at 1080p and a waste of money to go halfway and have the components but not enough horsepower to run them - thus the big market for upgrades.

Your savings going from 1080 to 720 will be much smaller, and likely only increase your frustration with too small a screen, over any small increase in CPU performance.

If you had a 4K laptop and kept is at 1080P, and only bumped it up for watching videos or playing games, you would have more CPU power available, but that's not the difference you are asking about (you only want to go from 1080 to 720). AND since you're not doing intensive work at the lower resolution the savings gained is even smaller.

I hope that was clearly explained.

As a personal example: I want my cellphone to operate quickly, to save power, and I have a tiny screen that can only display up to a certain resolution; I'm not going to be able to see over 4K if I wanted to.

I run it at 2K all the time (and with Android there's no auto-switching resolution based upon what you're doing). I get an average of 20 hours per charge and my phone benchmarks as the fastest. 

I don't need to go to 720p to squeeze more life out of the battery nor would I want the beautiful screen to operate at so low a resolution. 

With such a fast CPU (SD845) I don't worry about scraping a small increase in performance out of my phone (and in my case my gain would be greater than what you are asking about).