Is it safe to have my child vaccinated while I'm pregnant?

Yes. And it is definitely recommended unless you have other health issues. Please make sure your child's pediatrician is aware of any other issues you may have.
Depends on situation. I like to avoid giving your kids the living vaccines (mmr & varicella) during your first trimester out of caution rather than any known risk.The others including the flu are fine at any point in your pregnancy.