With the professional third party liability insurance: are my students covered if they injure and/or damage other people or their property?

If the injury/damage to the third party caused by the student is because of your mistake as an instructor, then the student is covered. If the incident to the third party is caused by the student’s mistake, because they did not respect your instructions or safety rules, for example, then the student is liable. They may be covered by the recreational third party liability insurance if they have it.