in urdu, persian, arabic, saraiki, punjabi, and pashto... the languages I know... there are usually only one word translation for both mouse and rat. and usually vice versa, yes there are more than one word for that type of creatures, but they are mostly considered as synonyms, maybe there are different explicit names for them in high circles of zoologists or lexicographers, but I am talking about daily usage of the word.
No, nothing to do with it. כ is sometimes pronounced as "k" and sometimes as "kh". There are certain rules when it changes its pronunciation, which are rather complex, but that's when it's the same root. In this case, these are two unrelated words, so the pronunciation is also arbitrary.