Equations: very yes. You have to create some custom short codes. Example of it in action here: https://lambdaland.org/posts/2024-08-06_zkp/

Here's the source for that entire post: https://gist.github.com/ashton314/e63c9ed31c38c421f11d84df1f...

The shortcodes are a little messy, I will admit, but I have a temp.el snippet that expands to the right thing so authoring isn't too bad.

org-babel: I'm not sure. Here's the docs for code block handling in ox-hugo: https://ox-hugo.scripter.co/doc/source-blocks/