When and why did the Romans stop calling themselves Roman and instead began to call themselves Italian? Also When and why did they stop referring to their own language as Latin?

There's definitely more room for further responses, but there's some good previous responses to similar questions which should provide a good start in understanding the answers to your questions, including:

/u/BaffledPlato answering the question When did people of the former Roman Empire cease to consider themselves "Roman?"
/u/HatMaster12 answering the question Why did a vernacular Italian language develop over a continued use of Latin?; and
/u/bitparity answering the question Did "Italians" in the early middle ages still think of themselves as Roman?

Hope that helps!