Short answer: no. The Sumerians said the Akkadians were north and west of them, not south. The only mention of people from what is now Arabia is Dilmun, which was a region in the Persian Gulf (likely Bahrain and Qatar). read more
No, they probably came from Upper Mesopotamia (around Mosul). This is where the first Akkadian inscriptions are found, and it is well known that they invaded Sumer from the north. Furthermore, inscriptions of languages closely related to Akkadian have been found in northeastern Syria. read more