How formidable were the Mitanni to ancient Egypt and other neighboring cultures? Were they a substantial threat, or a minor player in the over-all struggle for dominance of the region?

The Mitanni Empire was a short-lived super-nova in ancient Mesopotamia. Their capital city, Washukanni has not been located yet, so historical sources about this kingdom come from indirect sources, like the Amarna tablets in Egypt, Hittite documents from Hattusha and Assyrian sources from Ashur. There is also fragmentary sources about the Mitanni from the fringes of their empire. Notably the site of Nuzi, in Northern Iraq and Alalakh, in western Syria.
Saushatar seems to have been the most powerful king of Mitanni. He was able to loot bronze doors from a major temple in Ashur. However, it is hard to pin down the date Saushatar ruled the Mitanni kingdom. He ruled during a dark age in ancient Mesopotamian history and the historical sources for the Assyrians, Egyptians and Hittites are all silent during the period Mitanni rose to prominence. We don't know how many generations separate Saushatar from Tushratta, who is far and away the best documented king of Mitanni. However, it is probably no more than 100 years.
The military annals of the early phases of the Egyptian New Kingdom are replete with their numerous campaigns against the Mitanni kingdom. Tuthmosis III led twenty campaigns against the kingdom of Mitanni. Tuthmosis IV signed a peace treaty with this kingdom. They formed a marriage alliance with Egypt and the Mitanni king Artatama I married one of his daughters to Tuthmosis IV. Yet, this alliance with Egypt did not help the Mitanni very much when Suppiluliumas revived the power of the Hittite kingdom or when Ashur Ubalit revived Assyrian power. Eventually the Mitanni suffered from internal dissension and were caught between the expansionistic powers of the Hittites and the Assyrians. The Mittanni were a major player and substantial threat for more than 100 years, but only lasted as a major power for less than 200 years. Roughly 1500 BC to 1330 BC.
Primary Sources: "The Deeds of Suppiluliumas, as told by his son Mursilis II" in the Journal of Cuneiform Studies Vol. 6 1956

"the Tell El-Amarna Tablets" edited by Samuel Mercer 1939

"Ancient Records of Egypt. Vol II The 18th Dynasty" edited by James Breasted 1906