If I have an image of x/y resolution and display it on a monitor, does this mean that between a 15-inch monitor and a 42-inch, the same amount of pixels (x/y) will be displayed, only that each pixel will be of different size?
In relation to my first question, if I connect a 25-inch monitor or a 13-inch monitor to a computer, will the GPU/CPU work harder when connected to the 25 inch one? Could there be more lags on the bigger screen, or is it simply a matter of pixels' stretching?
The answer to both your questions is the same: the physical size of the monitor makes no difference at all. If your screen is 1920x1080 pixels, and you display a 960x540 image without any scaling, the image will stake up a quarter of the screen, and it won't take any more GPU time to display on a larger monitor.
The only caveat is that most OSes have some kind of pixel scaling method now, so that icons, buttons, and text aren't unusably small on modern high-resolution displays. In Windows the option is called "Change the size of text, apps and other items". If this setting isn't set to 100%, most things on the screen are scaled up even if the program isn't scaling them. This is more likely to be set high on a small, high-resolution screen than on a large, high-resolution screen. If you try to test displaying the same image on different-sized monitors, and they are using different values for this setting, the image probably won't be the same number of pixels on the monitors, because it's being scaled up according to this setting.
Ad 1) Yes, if both monitors have the same resolution but different screen sizes, this means the pixels of the larger monitor are bigger. It could also mean the space between the pixels is different.
Ad 2) No, the physical size of a monitor makes no difference for the GPU. It only cares about the resolution, for example 1920x1080 vs 3840x2160. The latter has 4 times the number of pixels, so it requires 4x the amount of memory and a simple operation like
glClear might take 4 times longer to complete. More complex operations might even be slower than that, due to their implementation details.
Regarding lag: I believe the time it takes to send the color information for each pixel is determined by a hardware standard and should be the same for each monitor. However, the monitor's chipset, which consumes that signal, may be faster or slower than a monitor of a different make or model. Same goes for the panel technology (e.g. TN, IPS, etc.), which might have a different refresh rate. But the hardware specifics have no effect on the GPU with one important exception: vertical sync. If vertical sync is enabled, your GPU will wait for a signal from your display before it starts sending the latest color frame. This will introduce latency, which can be pretty severe. There are solutions to that, but that's outside the scope of this answer. I recommend you to watch this video about it.