If I make a claim, will I have to pay medical expenses out of pocket and be reimbursed later?

It depends on the medical center. Most hospitals arrange direct payments with insurance companies. In this case, the insurance company will pay the amount of your eligible claim directly to the hospital.
With clinics it can be different, with most companies you will have to pay your bill for your consultation/treatment and make a claim afterwards. This is why it is very important to call the emergency service in advance to find out if your claim will be admissible.