JS Stack Overflow / Increase

Is there a way to ask the Interpreter how much stack space is left

I don't believe there is at the moment, no :(

Another question is how do fat arrow functions compare to standard functions?

They use slightly more storage sometimes because they store the value of this when they were executed. Otherwise there's not much in it.

Regarding minification, which is not directly stack / nested calling related but share the memory restrictions: does the Esprima minification take this into consideration?

I'm not sure I understand? It doesn't take anything Espruino-specific into account I'm afraid.

You can also try E.setFlags({pretokenise:true}) which means Espruino does minification itself - and compresses reserved words down to a single character.