Based on what I saw on the movie, I realized how surprised Count Dooku and maybe even Palpatine were when they saw Anakin using the Dark Side of the force. In the book it actually shows how Dooku was feeling at the time, and the curious think about his account is his surprise in his strength. Meaning the original plan was to essentially lure Anakin into defeat and force him to kill his own master, as in many cases of Sith inauguration into the ranks. As such, Dooku used his full strength too late and essentially lost his ability to become an effective Sith and leader.
There have been times where the Sith's rule of two have been broken due to what they would perceive the greater good, but many times because their is a rebellion in the Sith ranks. However, it is very feasible that Palpatines' original plan was to have two of the greatest members of the force underneath his guard. There is some hint to that Anakin was essentially a product of Sidious. Because of this, it is also probable that it was Sidious' plan all along to have his product by his side. If the theories are true.
After Dooku had failed and lost his hands (and his strongest ability as a Sith) it is no wonder Palpatine let Anakin kill him. I say "let" even though he was ordered too. There are two reasons for it. One, we all know that Anakin craved and longed for it, but also, Palpatine was able to absorb the guilt for him by not letting him make that decision. It brought the Dark Side a great ally, because, Anakin was seen has as a hero in front of the entire Republic, and as such, he knitted a stronger bond with Palpatine who actually let him express what was in his heart of hearts, instead of the restraints (at least what Anakin personally felt) of the Jedi Council.