Hooking a 30 Amp Shore Power system on a boat to a 15 or 20 Amp system is not a problem generally, as you are rated for that much current... but the adapter cords are a shock hazard and you really need to find a marina that uses proper shore power outlets.
As for the inverter... if it is properly wired, she should be able to use the hairdryer when on shore power with no problem, regardles of whether you are at a 15, 20 or 30 Amp shore power connection.
While I can't speak for using it on the batteries alone, I wouldn't use it based on the size of your house bank, which is only about 200 amp hours, and the 1500 W hair dryer will be drawing the equivalent of 125 Amps or so at the batteries when it is on... not a good idea... don't think your DC-side wiring is sized for it.
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You know what the first rule of sailing is? ...Love. You can learn all the math in the 'verse, but you take
a boat to the sea you don't love, she'll shake you off just as sure as the turning of the worlds. Love keeps
her going when she oughta fall down, tells you she's hurting 'fore she keens. Makes her a home.
—Cpt. Mal Reynolds, Serenity (edited)
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Still—DON'T READ THAT POST AGAIN.