Not just bands, but artists in general (staying away from classical, though... too easy)
France: Edith Piaf, Barbara, Jean Ferrat, Juliette Greco, Francoise Hardy, Camille, Yelle, Marie Laforet, Boris Vian, Chantal Goya, Jacques Dutronc, Serge Gainsbourg, Renaud, etc... Jacques Brel is Belgium, but made his name in France and is probably the most legendary (and best) French singer.
Italy: Katyna Ranieri, Nora Orlandi, Nico Fidenco, Piero Piccioni...
Russia: t.A.T.u, Glukoza, Planka, Victoria Dayneko... I think Vitas was born in Latvia, but he's made his name in Russia. Murat Nasyrov is from Kazakhstan, but started his career in Russia.
Ukraine: Nikita (all I know, coz I wanna fux Dasha Astafieva)
Germany: Marlene Dietrich, Kraftwerk, Reinhard Mey, Kurt Weill/Bertolt Brecht, Tangerine Dream, Hildegard Knef, Ute Lemper, Can, Faust, etc...
BQ: Tough, as there are many. My absolute favs would be Jacques Brel and Sigur Ros/Jonsi, but I'd also list Edith Piaf, Marlene Dietrich, Falco, Hakan Hellstrom, Jean Ferrat, Ewa Demarczyk, and Anna Jarvinen among my favorite artists.
BQ2: Probably either Third Eye Blind or early Sylvester (when it was Sylvester & the Hot Band and a cross between southern soul and glam rock, before the disco age).