If you are born mute, do you have to be deaf, or if you are born deaf, do you have to be mute?

No. Born mute, your ears will usually work perfectly well. The problem comes when someone is born deaf but with working 'speech organs'. Acquiring recognisable speech comes from hearing sounds and imitating them, and having the 'right' ones rewarded and reinforced. If you can't hear the sounds, you can't imitate them - or understand the response to your babble - if you even do. Even with partial hearing, words may be distorted as a child says what she thinks she has heard. (I say she because I know a young lady who can still lip read even though her hearing problem was virtually sorted out nearly 20 years ago. She taught herself to lip read, and this for quite some time masked her hearing problem!)

Nowadays, problems of deafness are more curable than they once were, and methods of teaching speech are vastly improved.

https://en.wikipedia.org/wiki/Hearing_impairment

I have met few deaf-mutes compared with the number of deaf or mute people I've encountered. Being deaf doesn't need to hold people back - https://en.wikipedia.org/wiki/Evelyn_Glennie - although she was 12 when she went deaf.