> What exactly is 3rd Order logic?
In short (and just a tad bit simplified), it's all about what kind of quantifications are you allowed to do:
First order: quantification over individual variables, i.e., variables representing individual objects the theory is talking about.
Second order: quantification over set variables, i.e., variables representing sets of objects the theory is talking about
Third order: quantification over variables representing sets of sets of objects the theory is talking about.
And so on.
> And what is going on with its relationship to elementary topoi?
I don't know. I have not looked too much into category theory.
> Previously I had believed that higher-order logics (above 2nd) require a semester of Model Theory.
Depends entirely on what you want to do with the logic. You will need knowledge of model theory for many topics related to first-order logic too (or any logic for that matter).
> Upon deeper investigation, to my surprise, I found that Model Theory is not used here.
Again, that entirely depends on what are you trying to do. You will not need model theory in every situation.
>Instead category theory and "elementary topoi" are used instead.
I guess whatever you're reading is building foundations starting from category theory, instead of the more classical approach of building on top of set theory.
> What the heck is going on here? I feel like I'm taking crazy pills.
What is going on where exactly? Seems like you got yourself confused or jumped to some conclusions.