Am I right about this?

One of the parents must have the mutated gene for the child to have a dominant mutation? Right? The mutation would be recessive if none of the parents had the mutated gene, am I correct?

In order for offspring to have a dominant mutation, they need at least one copy of the allele from either or both parents. So yes, at least one of the parents must have the mutated gene in order for the offspring to have the mutation.

For your second question, I'm not quite sure what you mean so I will restate two possibilities for what I think you mean:

1. The offspring does not have the mutation if none of the parents have the mutated gene: this is correct.

2. If the mutation is recessive and the child has only one copy of the mutated allele, then the child will not show the mutation. If none of the parents had the mutated gene, then it would not be possible for the child to have the mutated gene or show it.