Try QAPlug - Checkstyle plugin. Then go Analyze - Analyze Code (not the standard Analyze!) then click "..." near Run with chosen profile select a concrete profile and press the import button.
In the next dialog select the checkstyle rules file and then click the OK button. That's it. Just wait for the results. I got something like this.
That plugin has its limitations however. For example, I didn't manage to use a file with suppressions with it.
You can also use the plugin CheckStyle-IDEA which also has good configurations and allows you to use suppressions, custom checks and displays the inspection warnings in the editor immediately, but to apply that you need to set Project Settings - Inspections - CheckStyle - Real-Time Scan in the IDE settins.
Answering the second question, I didn't find any automatical reformat capabilities in these plugins, just checks.