There was Rudolf Hess's attempt at reconciling Germany and Great Britain before the invasion of Russia, which if successful, would prevent Germany having to fight on two fronts. I don't know if this really answers your question (Did Hitler... ), because this might not have been Hitler's idea, but here's the story anyway.

Firstly some background. Hess fought with distinction in the WWI and won the Iron Cross, second class, he joined the NSDAP on 1 July 1920, he was part of the Beer Hall Putsch, a failed Nazi coup, attempting to seize power in Munich in 1923, while in jail he helped Hitler with Mein Kampf and after the Nazis finally came to power in 1933, he was appointed deputy Führer to Hitler. All this to point out that Hess was actually a prominent Nazi figure and probably very close with the Führer, so this could be something Hitler knew of or even supported.

Now, on 10 May 1941 Hess took a Messerschmitt 110 and flew alone to Scotland, where he parachuted onto the grounds of the estate of the Duke of Hamilton, with whom he was on good terms. "The Führer does not want to defeat Britain" he first said to the duke and then to the officers sent to interrogate him. He told them that Germany would respect the British Empire in return for Germany being allowed a free hand in Europe. Naturally, this claim was not taken seriously by Churchill and was fervently and immediately denied by Hitler who proclaimed his former deputy insane.

The truth of the situation will never be known as Hess took it to his grave after serving a life sentence in Spandau prison. Was it the act of an envious leader who felt sidelined and overtaken by Goering, Goebbels or some other high ranking Nazi officer (the explanation offered by William Shirer)? Or was he sent on the mission with Hitler's blessing, in the hope of making peace with the British so that Germany could attack the Soviet Union unmolested?

I hope this helps!

Source: Adolf Hitler: Evolution of a Dictator by Luciano and Simonetta Garibaldi

Hitler offered peace to the United Kingdom following the fall of France in June 1940. In a speech to the Reichstag on July 19th 1940 he stated that he saw no reason that the war should go on.

In making this offer Hitler hoped that his western front would be secured and he could turn to planning the invasion of the Soviet Union in peace. If the offer were rejected, which obviously it was, then he could again tell the German public that he had tried to be reasonable but that the UK wanted war.

Also, the main goal of the Ardennes offensive in December 1944, known as the Battle of the Bulge, was to inflict such a massive defeat on the western allies that they would be forced to seek a separate peace. Again leaving him free to focus on the Soviets.

Sources :

The Rise and Fall of the Third Reich by William Schirer

The Guns at Last Light by Rick Atkinson

Not sure about your second question. Hitler never had any intention of surrendering and even if the Allies had negotiated a peace settlement (also extremely unlikely) handing over Hitler for trial would have been the top condition.