Wouldn't local cops violating people's constitutional rights put them at odds with federal law?

Yes, but that just means prosecuting individual cases. That is very different from trying to make systemic changes in the way local cops operate; those kinds of changes must be made by local governments, and are the kind of changes local citizens should be demanding, and should be voting local governments out of office for not making.

Yes, and as a general rule when cases are brought they are under the auspices of a federal civil rights violation. These are very difficult cases to prosecute though, so I think the FBI overfocuses on lower hanging fruit unless there's a political motivation from the top to drive this.