The photo was published (and mis-identified Alexandra) in an exhibitioncatalog. It is definitely not her - I too think it is Olga (due to the nose; Tatiana was more slender).
I read that only the eldest daughter had a considerable gift for music - the others seem to have played "quite well" but nothing more. Alexandra - as a highly gifted and trained pianist (inherited from her mother and grandfather who were both brilliant players) definitely had some lessons with her children too but of course there was no time to teach them frequently (I think it is necessary to get lessons from a "stranger" - because if only "Mama" teaches one, the little Ones are not as serious as they have to

Alexandra - as you said - loved Wagner and played most of his operas (even for Queen Victoria). She loved Brahms who frequently visited Darmstadt in stayed in contact with her mother Alice and her private scretary Dr. Becker (whose daughter Antonie "Toni" became one of Alix's closest friends). She also played Beethoven sonatas and much much more I presume (look at the incountable music-books in the photograph with Olga).
Mabye some are preserved at GARF or somewhere?