In Qatar, it is mandatory for a car to have insurance cover always. That is the law. The only time when a car can run on the road without insurance cover, is when the owner has not renewed his Istimara and also his car insurance. In order to get an Istimara renewed, documentary proof of valid insurance for a minimum one year, is a must.
Now coming to the queries posted. The 3 quotations are required to allow the authorities to estimate the cost of repairs. As such, what the garage people are saying is not correct. As for the second query, yes you can go ahead and repair your car once you have the Police Report in hand as you need the car for your use.
Coming to Angelo's question, if the police is calling you, they are not joking. They have a situation in hand which they are trying to resolve. While, they are calling you, I believe they are also calling the other person and pressurizing him to pay for the damages. Once he does so, the money will be handed over to you. The other person's car may then get released by the police with or without a penalty.