Question: Alexander shoots the moose with a rather rapid fire action: it sounds as if the rifle is semi-automatic. What was the true story, and why did Alexander specifically ask for a a cal .22 rifle? Was it because he planned to hunt small game, and knew that a heavier rifle/calibre would be unsuitable? To plan for moose hunting, he would choose something like cal.308.
Question: Chris didn't take any ID or such with him. It's OK that he could work with the country guys, but how could he get an "official" job in a fast food restaurant without any identification document or card?
Answer: Chris probably knew what his Social Security Number was, and providing this could be enough to legitimately get him on the payroll. The people he worked for along the way could have been used as references. It is also possible that he was able to get a transcript and/or other documentation from Emory University; maybe a phone call to check if he graduated was more than enough. Christopher was also somewhat older and educated/ intelligent than others seeking such employment - in his early 20s compared to high school kids - which may have given him an advantage. There was probably a high job turnover rate among the employees, making it easier to get hired. Fast food restaurants tend to hire people from all walks of life, many who do not have much, if any, formal education or prior job experience; they often hire whoever applies.
Not entirely disagreeing with your answer, but having worked at a university, I can say that someone cannot simply make a phone call to obtain a student's academic information, even their own. A 1974 U.S. federal law (FERPA) protects student privacy. Every school is different, but there is usually a process requiring identification, paperwork, and signatures to prove identity. As Chris had left all his I.D. behind, it would take some time for him to get any college information, particularly from a school in another state.
Answer: In real life, his sister Carine recovered his backpack, which had been taken and kept by a man immediately after Chris died. In it was his wallet, along with his SS card and other important documents.
Answer: One possibility is that it wasn't an "official" job and his boss was paying him 'under the table' (unreported employment). It is illegal, but it's more profitable for an employer to avoid reporting anything to the federal government, disregard regulations, not pay the usual employee taxes, benefits, etc.
Answer: Having read Krakauers book: The rifle was a .22 Remington scoped semiautomatic. Accounts at the time mention that this may not have been the best choice, and it is clear that McCandless did not have enough experience to live in the wild, although it is impressing enough that he managed to survive for 16 weeks.