Hitler being who he was could not have backed down in the face of Stalin's threats. It would have meant a huge loss of face, not just internationally but with his own generals. If the Czechs publicly declare they will not yield any territory and will fight if invaded I think Hitler would still choose war. In OTL Case Green remained in effect even after the Czechs began mobilization and France began a partial mobilization and even Chamberlain stated that if France went to war Britain would feel obliged to do the same.
Given that Hitler will not back down in the face of threats from the hated communists.
Chamberlain would definitely see Stalin as escalating a delicate situation and blame him rather than Hitler. Remember that at THIS point, mid 1938, not only is appeasement the current policy but Hitler appears to only be trying to bring German nationals back into the Reich. His demands did not seem all that unreasonable and there was a sense that the Versailles Treaty had been unjust and modifications were in order. In Chamberlain's point of view Hitler's demands were reasonable and the problem was finding a method to placate him peacefully. To the Prime Minister Stalin's rash guarantee and the Czechs refusal to negotiate would be the problem.
If Romania was willing to allow overflights and the passage of Soviet troops I think France would feel obligated to fight if their allies Czechoslovakia and the USSR got into a shooting war. Even if the UK stayed out of it Germany would be doomed given how militarily weak they were at this time.
Hitler was never one to acknowledge military reality and I believe he would insist on Case Green no matter what. The result is either his arrest by the generals and a coup, or a very short war and quick defeat. Either way Hitler and the Nazis don't survive.