> 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.