1. Is the following correct usage? "They didn't want me to ruin the family name. "
  2. What are some other idiomatic expressions or phrases to say something similar? I am aware of "give a bad name" and "one's name is mud".
  3. What would be the correct usage of idiom " one's name is mud" in this context?

Solution 1:

Your usage in line 1

They didn't want me to ruin the family name

is perfect. And as you say,

to give the family a bad name

would be fine too.

It's funny to see One's name is mud: a wonderfully concise example of bathos! An idiomatic use of the expression would be,

They were worried that (if I did such and such) the family name would be mud.

Other expressions:

They were worried that (if I did such and such) the family's reputation/standing would/might (thereby!) be ruined.

Or

They didn't want me dragging the family name through the dirt

Or

...miring the family name / besmirching it / bringing/heaping dishonour upon/on it.

Or, more plainly,

They didn't want me to give the family a bad name.