"I think 'after doing something' and 'after to do something' are both correct"... no, they're not. This kind of thinking (ie. assuming) sabotages your progress. Just ask for advice. "She attended university and completed her doctor's degree after stopping playing table tennis." Correct, but awkward. Native speakers would not say or write this. "She attended university and completed her doctor's degree after she stopped playing table tennis." Correct and natural.

"after doing/stopping etc" = "after [subject] did/stopped etc e.g. After finishing / After I finished the book, I started cooking dinner. "after [subject] doing/shopping etc" is wrong. "after [subject] to do/to shop etc" is wrong.

It should be replaced with "stopped" instead. The "after" used here is a result of or because of something that has happened so it shouldn't be "stopping"

Great! After is used in the following ways: as a preposition (followed by a noun): I went for a swim after breakfast. as an adverb (without a following noun): He died on June 3rd and was buried the day after. as a conjunction (connecting two clauses): After you’d left, I got a phone call from Stuart.at a later time a. when a particular time has passed, or when an event or action has ended After the war, my parents moved to the U.S. Essays handed in after 12:00 on Friday will not be accepted. She is leaving the school after 20 years as its principal. This message arrived after everyone had gone home. after a while: It seems noisy at first, but after a while you get used to it. after doing something: Wash your hands after touching raw meat. Please pay attention to the words used in the above mentioned sentences. After to do is wrong.