You have interesting beliefs about the level of knowledge the chairman of the JCS would have access to. He wasn’t ‘conducting talks’ with a foreign government, he was reported to be reassuring them that, essentially, a temporary internal crisis would remain, well, temporary and internal. I’m curious why you think that on his way out the door Trump would have been engaged in some major diplomacy with China.Absolutely no. He has no idea about the current diplomatic talks between the 2 countries. He could derail very important talks. His sole role is advisor to the president, he has no command authority whatsoever. He can't conduct talks with a foreign country no matter what.
Is this common for military officers to go around the authority of the commander in chief and make decisions that could affect foreign relations? If he says the wrong thing to his Chinese counterpart it could really be catastrophic. He is not a diplomat he is a soldier.
Of course this is not a usual thing, but neither were they usual circumstances. There was a lame duck president on the way out the door who by many reports was pretty desperate to pull any strings or levers he could to reverse the outcome of the election and to stay in power. That’s not a usual or particularly safe set of circumstances for the most powerful nation in the world.