Basically, Peter III of Russia was a moron. He neglected his wife, he wished that he could rule a more civilized country like Sweden, and he barely spoke any Russian. He
dropped out of the war, essentially a slap in the face to the Russians who had enlisted and lost their lives fighting the war, because he thought that Frederick the Great was cool and he admired him. If you want Russia to stay in the war you need to either keep Elizabeth alive or have her name someone else as heir. This might be bad in the long run, though, because Catherine the Great will never come to power.
Practically all of the above is taken from the official history promoted by CII and, as such, is mostly wrong.
To start with, "neglecting wife" was more or less expected from an aristocrat and it also should be noticed that his wife actively cheated on him. Wishing to be a king of the civilized country is not an indication of being a moron, rather it is other way around. Degree of his grasp of the Russian we can't define now but it seems that he was freely communicating with his subjects. OTOH, Catherine II, with her alleged "perfect" Russian, hold a record of making 4 mistakes in 3 letter word.
Dropping out of the expensive and rather idiotic war in which Russia did not have any clear interest (read protocol of the Conference, March 15, 1756) was the right thing to do and it is worth noticing that most of the unhappy noises had been coming (a) from Bolotov, author of the memoirs, who comfortably spent most of it in Konigsberg, (b) the Guards who spent it in St-Petersburg, and (c) Catherine herself. Needless to say, that as the Grand Duchess Catherine (her father was
Prussian general) was in a complete agreement with her husband and was taking money from the British Ambassador (the "Young Court" was almost openly pro-Prussian). After the coup she did not change a word in the agreement and the Old Fritz remained her closest confidant all the way to 1771/72 when he forced her to agree to the 1st Partition.
OTOH, there was no unhappiness related to the peace from the people who were
really involved in the war like Rumiantsev, Buturlin, etc. Anyway, by the time Peter signed the peace Russian-Austrian relations had been almost completely broken with the Russians blaming Austrians for failing to supply them, trying to win a war by the Russian hands and other sins and the Austrians being openly unhappy as well.
The Old Fritz was not necessary "cool" but he definitely was the greatest general of that time and a very capable monarch who arranged an orderly and efficient administration of his kingdom. Small wonder that a ruler familiar with the Russian mess wanted to follow the good example. Actually, during his short reign PIII pushed through a number of the important laws credit for some of which was post-factum appropriated by his widow.
CII not coming to power would not be necessarily bad for Russia because, while being superficially "glorious", her reign was truly remarkable for its wastefulness, corruption, inefficiency, foolish foreign policies (making her former boyfriend a King of the PLC was just a beginning) and absolutely terrible domestic policies which had a disastrous impact upon the Russian economy: she turned serfdom into a slavery thus killing for almost a century possibility of the industrial development. Being absolutely unlawful monarch, she had to keep pleasing the Russian nobility just as a way to survive and to please those close to the throne (and more dangerous because they could support her son) by the enormous gifts of money, jewels and lands with the serfs (during her reign 850,000 "state" peasants had been given to to the provate owners