Are space-time dimensions a fundamental property of the universe or an emergent result of other physical laws?
The number of spacetime dimensions is set, as far as I know, empirically through observation, although there have been studies arguing this exact question. For instance, I recommend you to have a look at the paper Is the (3+1)−d nature of the universe a thermodynamic necessity?.
There are also studies arguing that the actual number of dimensions of our spacetime is greater than $3+1$. I'm referring to string theories, in which consistency conditions specify the number of dimensions. There is an interesting discussion about this (although very mathematical) here. However, string theorists then use the tool of compactifications to return to a $3+1$-dimensional spacetime.
So, regarding your question, I don't think anybody has ever considered whether the existence of spacetime dimensions is a fundamental property or an emergent result, while the nature of the number of them has attracted attention and there exist theories arguing that it is a fundamental property and theories arguing that it is an emergent phenomenon.
However, I recommend you to also read the topics mentioned by @Qmechanic.