Can I play while my phone is charging?

Yes.  Do note that if you are charging your phone by connecting it to your computer and experience difficulties, check that your phone's USB connection is not set to "USB Mass Storage". To verify and change this setting, just drag the top Status Bar down and select "USB Connection".  Next, click the "Charge only" option and tap "OK".  Different phones might have different methods of disabling this feature.  Please check your phone manual for details.

Please note that you must ensure that your SD Card is "Mounted" while you are not charging your phone. To mount your SD Card, go to Settings > Storage (or SD card & phone storage) > Mount SD Card.