No, it does not flush
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If the water is entering the toilet from the bowl then the water is backing up from whatever appliance or fixture is being used at the time.
Other than that the problem you are having is the flapper inside of your toilet. It is allowing water to leak past it. These are easily replaceable and inexpensive. You may wish to try cleaning the edge of the tank where the flapper seats against the tank of any debris. Checking the sealing surface of the flapper might help too.
As for the clogged toilet I think that it would be best if you had someone come out and clear the lines for you. They will be able to determine if there is a more serious problem with your sewer system.
Best regards, XXXXX XXXXX Smithy
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You can stop the water from entering the tank by turning off the valve at the wall where the water line goes into the tank.
You need to determine what is still letting water into your sewer lines. It could be another toilet, a leaking faucet or anything else that has water supplied to it.
As a last resort untill you can get a plumber to unclog your lines you can turn the water off to the whole house and then open an outside faucet nearest the lowest point in the water system . This will stop any more water from being added to the sewer lines.