In Answer to your Questions,
Melian married Elu Thigol/Elwe who was the cousin of Olwe, who was the father of Earwen, Who was the wife of Finarfin, father of Galadriel. So she was Melians Husband's, cousin's, Daughter's, Daughter.
The Power that Melian had was the power that to Maia, though they did not all have the same type of power. Melian used it to defent Doriath with the girdle of Melian. Galadriel however used the ring, Nenya, one of the three, to guard Lorien from time and decay and evil.
Something was given to Galadriel by Gandalf/Mithrandir when he came from over the sea according to the unfinished tales. It was the Elessar that was later in turn given to Aragorn. But this was not the source of the power nor was it supposed to have been given by Melian.
whew...Mouthful. I hope this answers your question.
P.S. Galadriel did spend most of the First age in Doriath With Thingol and Melian. [img]smilies/smile.gif[/img]
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Instead of pepper spray, you pack a glass bottle and scream "Aiya Earendil Elenion Ancalima!!!" at muggers.
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