She wouldn't know a thing about being a thief, plus she still has a job as police inspector, so if anyone from interpol found out that she was working for a band of thieves, then history will repeat itself from Sly 2.
Yep. Carmelita would never intentionally break the law. If necessary, she would find a legal alternative (say something needed to be stolen to bring Sly back to the present, she would look for a way to obtain it through her INTERPOL resources, rather than being aware or part of a heist).
She could continue chasing the Cooper Gang involving minor cases or thefts (e.g valuable stamps, chocolate bars, watches, ect), while in major cases involving the plans of crime lords, she would listen to Bentley and/or Murray and provide undercover help for them.
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