yes, she is right
게 means a thing
수진씨가 좋아하는 게 뭐예요? What is 수진's favorite thing? you can also say 수진씨가 좋아하는 것이 뭐에요?
yes, 게 is a shorten form of 것이. 게 is preferred only in spoken Korean.
끝말잇기 is a game connecting words by the last letter.
If you say '호랑이'
Next person should say word starting with 이 like a 이빨
And next person should go like 빨래
and If now it's your turn and you don't know any words starting with 래, you are losing the game.
and in most common 끝말잇기, the proper noun or name is not available. because it will never end then.