As of 1 January 2018, an amendment to Act No. 48/1997 Coll. states that if you are studying in your first doctoral programme in full-time form, health insurance is paid for doctoral candidates by the state — but only if you do not have an employment contract. If you do have an employment contract, health insurance is paid through that employment.
If you are over 26, you are obliged to notify your health insurance company that the state pays your insurance due to doctoral studies. UCT Prague should report this on your behalf, but it is advisable to verify this with your faculty dean’s office or directly with your health insurance company.
If you terminate your employment at UCT Prague (or elsewhere) and do not have another one, you must notify your health insurance company. The insurance will then (provided the conditions in the first paragraph are met) again be paid by the state.
By the way, most health insurance companies now offer their insured persons electronic access to their records (similar to online banking), where you can check what data and payments the insurance company currently records for you.
Social insurance is not mandatory; you do not have to pay it (though voluntary payment is possible), nor do you have to report anything. Social insurance will be paid if you have an employment contract.