For example, I have two C libraries: musl and glibc. For my situation, I need speed, thus I want to use the library that would "run the fastest". Would this involve cataloging the time and space complexity of each function per library? Is there an objective way to determine which library would have a worst lower time and space complexity?
Determining the fastest library for you is done through benchmarking, meaning timing how long it takes with realistic samples.
Specifically, determining time and space complexity of an algorithm won't tell you how fast it is. It's a measure of how its performance scales.