I want to get a 13" 144hz 1440p IPS freesync monitor, for a desktop pc. Benchmarks show, on a regular size monitor, my CPU will be able to put out 60 fps in 1080p, at 60hz. (a.) Will playing on a smaller screen allow me to be able to get my money's worth out of a 100-144hz screen, (b.) and also at 1440p, since my screen is smaller.
I'm assuming you mean a 31" monitor. Your computer and graphics card's performance is completely agnostic to the actual size of the monitor. Only does the resolution of the monitor affect performance. Higher resolution means more pixels which means more power needed to render the image. If you post your actual model of processor and GPU, people would be able to get a better idea of how your computer would perform on a 1440p display.
Presuming 60fps @ 1080p, you could expect around 40fps at 1440p. And that is being a little generous. If you want a high refresh rate monitor, I would recommend getting a 144Hz 1080p display. There are a good amount of economical options now that it's 2019. It doesn't sound like your computer is capable for 1440p.