What would have happened if Mussolini agreed to surrender and work with the Allies in either 1942 or 1943? Would there have been any backlash from Hitler? Could Italian troops have taken Southern Germany at least and end the war earlier?
No chance of Mussolini agreeing to work with the Allies. But seeing the German fails in North Africa and Russia, and the roaring success of the USA in the Pacific, he could take the opportunity to accept a coup and flee, leaving his successor to appeal to the Wallies. The difference from OTL is that Mussolini's exile is facilitated by the new Italian gov't, so he does hang so quickly.
After the invasion of Sicily IIrc Antonescu suggested that Italy, Romania, Finland, Hungary and Bulgaria all leave the war together. It was the last real chance for the minor Axis powers. If Mussolini wouldn't take that opening, it's hard to see him ever turning on Hitler.
If Mussolini had joined the allies in 42 or 43, Hitler would attack Italy The specifics would depend on when the betrayal happened