I tend to think of the communication at the end of the Avengers as kind of a unique situation, Clint has several people to deal with who don't follow proper protocol, wouldn't understand SOP. Rogers might, but he would be outdated, so it's easier to speak plain to Cap, Tony and Thor, rather than explain what A-O's, enfilades and defilades and choke points are on the fly. Or it's just bad writing.
I also always saw the comment in Thor to Coulson as more a dig at Coulson, because the first part of the exchange never gets mentioned with the "are you sending in more guys for him to beat up?" Especially if you read the comics that kind of fills in the behind the scenes action, Barton doesn't come off as Coulson's subordinate to me but rather Fury's handpicked guy saddled with a suit, he is the one who says they should cut Thor loose to tail him while Coulson wants to keep him locked up. I do think, he finds Phil somewhat amusing, but I don't get the vibe of camaraderie between them. This Hawkeye seems more Fury's beast to me, I mean the way they talk to each other at the beginning of the movie, it seems like Fury allows Clint a fair amount of slack.
The "what are you doing?" line, I think is kind of face value for me, it's shock. I think too much is read into it for a throwaway line in the middle of an otherworldly firefight where the guy is trying to process a thousand different things at once.