Question: Before the Clone Wars, what situations did Jedi handle most of the time? I only ask because at Padme's apartment, Obi-wan says it's not their job to investigate the assassination attempt. Anakin says that local security could guard the apartment; it's not a matter for Jedi. So apparently, Jedi don't watch over people nor investigate crimes. What do they do?
Answer: Their job is to protect the integrity of the republic with diplomacy and advice to politicians. They do not enforce laws or rules. They can also be security for senators or be taken to places in crisis to help resolve difficult matters (like the blockade of Naboo). They protect and secure. That was their job during the time of the 'old' republic.
So, in a way, are the Jedi similar to involving the FBI in a situation, instead of contacting your town/city/county police?
Maybe more like UN Peacekeepers.
Answer: I don't think that Dooku, nor Palpatine, really cared much about being actual Sith Lords. They wanted to be rulers in their own rights. Hence why Palpatine becomes Emperor Palpatine and abandons the "Darth Sidious" title. Dooku might have "destroyed the Sith" in the sense that he would kill Darth Sidious/Palpatine with Obi-wan's help.