score:3
A couple modern authors note that his politics were a strain on his career, such as this passage from Fighting Elites: A History of U.S. Special Forces by John C. Fredriksen (p101):
His seemingly procommunist sympathies resulted in trouble with superiors, but following American entry into World War II, Carlson was allowed to command the elite 2nd Marine Raider Battalion.
More specifically, a couple of passages from the book American Commando: Evans Carlson, His WWII Marine Raiders, and America's First Special Forces Mission by John F. Wukovits speak to the hesitation of the Marines to even allow him to re-enlist (p17):
A military observer, Marine Lt. Col. David D. Barrett, on station in the office of the military attachΓ© in China, wrote his superiors that communist officials had met with Carlson, "formerly a major in the United States Marine Corps, whom they consider their staunch friend". The more deeply Carlson became embroiled in China affairs, the more likely he was to be labeled a communist by other Marine Officers.
And (p18):
With events rapidly spinning out of control in the Pacific, Carlson reapplied for a commission in the Marines, a move that outraged fellow officers. He had already left the Corps once - a traitorous act to some - and he returned tainted by communist poison. How, they wondered, could he be an asset to the Marines?
Lt. Col. Merritt A. Edson, Commander of the First Battalion, Fifth Marines, raised the most strenuous objections. Like Carlson, Edson had served in Nicaragua and studied guerrilla operations, but he detested Carlson's association with the Chinese communist military forces. Edson, a blunt, brilliant officer, had little use for what he saw as Carlson's idealist, simplistic views of society.
Even after his death he was named in investigations into the influence of communism in American government and the military. For example, his name appears on a list of subjects for investigation in the investigation report on the Institute of Pacific Relations by Senator Pat McCarran in 1952. Its categorization is as follows (p152):
Identified as a member of the Communist Party by one or more duly sworn witnesses.
Made one or more trips to Communist territory.
Deceased.
Affiliated with: Amerasia (exhibit 1355). Committee for a Democratic Far Eastern Policy (pp. 4610-4611).
So in a word, yes - he was suspected of being a communist and was unpopular with the establishment. I'd check the linked material as a start for answers on the rest of the question.