What is the difference between kaç and ne kadar ?
"kaç' is 'how many.' 'ne kadar' is 'how much' :)
I don't suppose bütçe is derived from a French pronunciation of budget, is it?
Look here: https://en.wiktionary.org/wiki/Category:Turkish_terms_derived_from_French
How big is our budget?
The english translation is very awkward (native American english speaker). We would say either "what is our budget?" or "how much is in our budget?" - "How big is our budget?" would be a different sentence.