Do higher resolutions play a role in stuttering caused by low CPU performance or lack of cores/threads? If so and the laptop were lower resolution, I can see why a laptop I5 could perform better than a desktop I3.
Not very much. Resolution is pretty much exclusively the domain of the GPU since the CPU mostly sets up the scene and sends it over to the GPU and the GPU does all the 'turn this scene in to a frame for the monitor to display' work . . . it's more complicated than that on what the CPU and GPU do, but for the purpose of the question the answer is pretty much no.
This is the reason you see many benchmarking sites will simply lower the resolution when testing CPU comparisons in games because it will help remove the GPU as a limiting factor in the test since the GPU has less work to do but the CPU still tries to play the game as fast as it can.