As far as Mace Windu's conduct near the end of the duel, I think he was acting properly. It is true that normally once opponents are clearly defeated that the Jedi would spare the lives of these opponents (for the moment) and the Jedi would then turn the criminals over to the normal Republic civil authorities for trial for their crimes.
But this was a special circumstance. Palpatine had some officials in his pocket, either directly or through his alter ego as Sidious, he had corrupted some parts of the senate and had a certain degree of control over the courts, as well, so that he could effectively corrupt the judicial process. Meaning there was no way to be sure he would be convicted if he went to trial, even though he was guilty as sin. Not to mention the risk that he would use his henchmen to escape from prison before his trial could begin and flee from Coruscant only to go on with his efforts to destabilize the Republic from afar. Or run back to the Chancelor's office, trying to take back control of the Republic, all the while claiming that it was all some plot by the Jedi to take control of the government.
Under those circumstances, considering the number of people that Palpatine had murdered in the war he caused, the fact that he was the sith lord they had been searching for all this time and they finally had him, and that there was a very real possibility that he would escape justice yet again if he were turned over for trial due to his corruption and control of the government, Mace Windu had little choice but to kill him right there. Sometimes it is simply necessary to take swift and decisive action to defeat evil, and this was one of those times when it was necessary to do that. It's a shame, really, the whole thing could have been over with right there. Palpatine's identity as the sith lord and his evil plans fully exposed, his corrupt henchmen dealt with, the war over and the Republic restored to peace, but for Anakin's weakness in resisting the dark side. But eventually, after a long struggle for 20 years, good did triumph over evil.