A failing alternator can trigger the charging light (battery symbol) to turn ON but not the CEL. Therefore, you may get new insights into your problems by retrieving the CEL code(s) as described here:
http://www.honda-tech.com/zerothread/1901557
After you retrieve the code, note it and then clear the ECU. If the CEL turns ON again, retrieve the code to verify that it is the same as before. Troubleshoot any repeated fault code and make appropriate repairs.