The pink paper you were handed upon release is actually your drivers license until there is an outcome at the DMV hearing. It is a full license allowing you to drive anywhere you want and can be presented as an ID. It may be a good idea to also carry a photo ID such as a passport or other government issued identification. If you do not have a picture ID go to any DMV and request a California Identification Card.
The pink paper will remain good until the DMV sends you another temp license. It may take more than 30 days to get the subsequent temp but the pink will remain valid because the attorney requested a stay of the suspension, meaning the department will keep your driving privilege valid pending a full formal hearing.
Can I rent a car with the pink license? Sure, just be sure to also have a picture ID with you. The car rental companies all do a DMV check anyway so they confirm your license is valid via the computer database.
Should I notify my insurance of the incident? No, unless there was also an accident that caused damage or some other loss there is no good reason to call your insurance company at this point. A dui arrest is simply an accusation, there is no conviction until you are adjudged guilty in court or after a DMV hearing. Therefore it's nobody's business but your own. Do not call your insurer to request an SR22 unless specifically advised to do so by your attorney.