Can a player attach a downgrade to their own character? I've had my eye on using the dynamic with 0-0-0 and the downgrade shadowed. Shadowed reads "after attached character uses its action (including Power Action) you may deal 1 indirect damage to its controller". this would then allow 0-0-0 to ping a indirect at the opponent everytime the characters action is used.

No. Downgrades must be played on opponent's characters, just like upgrades must be played on your own.