Can I do sit-ups after herniated disk surgery?

Yes. Most exercises are good but certain ones should be avoided until you are free of symptoms like rowing and certain types of weight lifting. Even running may hurt in certain instances but then you can cross train. Core exercises are good as are cardiovascular ones but you can learn these from a physical therapist or by a spine specialist . If in no pain, regular exercise minimizes recurrence risk.
Depends. Many variables needs to be considered including your status of overall fitness, location of your disk surgery, level of involvement and the nature / complexity of your surgery. Make sure you have a thorough discussion with your surgeon to minimize suprises and/or disappointment postop.