문장 앞 with와 but on how 질문드려요

“With Korean food offered in a traditional Japanese pub environment, most customers mistake it for Japanese.”에서 문장 앞 with의 쓰임을 모르겠어요.
그리고 “Happiness depends not on what happens, but on how we handle what happens.”에서 but on how를 “how we handle on what happens”로 바꿔 쓸 수 있나요?