Both questions are due to the flapper. Remove the tank lid and look at the flapper. It may have a broken connector (small ears) where it attaches to the overflow tube or it may have mineral deposits on the seating area under the flapper. Turn off the water supply valve under the toilet. Flush. Now lift the flapper and clean off the seating surface and the bottom of the flapper to remove deposits. Drop the flapper back down and adjust the chain/strap that connects to the flush handle. There should be only enough slack to allow the flapper to seal. Turn the water valve back on and let the tank fill. Flush. The flapper should stay up until most of the water is out of the tank. The tank should then refill and not allow water to leak through to the bowl. If it still leaks through, purchase a new flapper and install it. As a note, take the old one with you and get the same type. Many new types are available with promises that can mess things up more than solve problems. Good luck.
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