You have four questions.
This made me wonder, do the laws of war say anything about misinformation or psychological warfare?
Yes, a little. Mostly in the form of specific prohibitions.
Is it e.g. a war crime to claim that the political or military leaders of the country you are at war with have ordered their troops to surrender, in order to mislead those troops?
No, not as far as I could find.
Is it a war crime to lie about the actions and declared intentions of your enemy to motivate your own troops?
No, as far as I could find.
Is there in fact anything that binds a country at war to factual representation of their own and their enemy's actions, declarations, capabilities and intentions?
Yes, but only in a very limited context; indeed, misinformation is explicitly allowed, under the 1977 Protocol I Additional to the Geneva Conventions of 12 August 1949, Article 37, Section 2, which states that "ruses of war" that do not rely upon the protections of the Convention to succeed, including "he use of camouflage, decoys, mock operations and misinformation" are explicitly allowed.
Some prohibited ruses include feigning protected status(non-combatant, surrendering, neutral party, United Nations, sick/wounded), feigning neutral status*, or feigning being part of the enemy*. (Source, ibid, Article 39).
*Except when conducted as part of naval combat operations(which have their own traditions, and generally allow pretending to be the enemy for purposes of transit, escape, etc. but require to fly one's true colors before taking offensive actions, as well as honoring surrender), and espionage (which is generally treated as a "crime" rather than an "act of war" under international law, when one is not wearing the uniform of one's own forces).