Adapter cords would only be a shock hazard if they are defective. All outlets have to be properly fused ie a 15 amp outlet has to have a 15 amp fuse or breaker a 20 amp outlet a 20 amp breaker etc. Therefore you cannot seriously overload a circuit. I say seriously because circiuts should only be loaded to 80% of the circuit capacity, but a 15 amp breaker will not trip at 80% of the rated load. The loads you wish to put on will be fine provided you don't have significant other loads if your wife uses her hair dryer, which draws 12.25 amps.
As far as using the hair dryer with a 1000 watt inverter, it does not have the capacity to run a 1500 watt load for anything more than a momentary surge current. Do not.