Probably not. Like most public agencies, police departments are often woefully underfunded. Property and financial crimes tend to take a backseat to violent crimes and crimes that put public safety at risk.
With that said, police are more likely to investigate in situations where you're able to provide them with the identity of the person who stole your information and when that person lives in the jurisdiction over which they have enforcement authority. If you have documentary evidence inculpating the identity thief, that makes it even more likely they'll attempt to investigate and prosecute.
Absolutely! Regardless of whether or not law enforcement looks into the circumstances surrounding the theft of your information, the police report itself has a great deal of value. As I discuss in the page on the FCRA, a credit bureau is required to block the fraudulent information if you provide the bureau with an identity theft report and indicate that you didn't authorize the subject account or transaction. For the most part, the credit agencies have interpreted identity theft report to mean police report.
Additionally, providing a police report to a creditor or debt collector is the quickest way to get them to leave you alone about the account. So not only should file the report, it should be one of the first steps you take.