Are there any protests about things (real, important things) that the Russian people disagree with and want their government to do differently?
Nah, only the usual European things... gay rights and such. [/trolling]
It seems ironic that the communists were so sensitive, like royalty. This sensitivity is part of what led to their downfall. There was also the financial incompetence, which could have led to some kind of inferiority complex... But, as you can see, trying to silence criticism and opposition was a fatal mistake.
Actually, you are not too far from truth there. They were communists in name only. They de facto continued the system of monarchy, only with a much stronger oppression apparatus.
Monarchy or dictatorship can be good, if a monarch does not only think of his pocket. Alas, such monarchs are far between.