Has anyone won a tennis Grand Slam without being broken in a service game?

Answer 1

Not in the literal sense of your question: No.

"A tennis Grand Slam" requires that all four majors be won in the same calendar year.

There have been not more than a handful that have ever achieved that feat. None of these would have gone through all four events without dropping a service game somewhere ... so the answer would be a certain: 'No'.

However, if you are to mean getting through all the rounds of a single Grand Slam event without dropping a serve anywhere, then I still doubt it, but it could certainly have been possible during the event's first 20 or 30 years, as back in the 1800's, players rarely travelled to compete outside their own national countries. All competition was amateur; travel was slow and people had jobs to hold down.