Book case is 本棚 (Hondana)
Bed is ベッド (Beddo) which is based on the English word. Also to note that western-style beds are rare in Japan. Futons (which are much different than the American version) are more common for the house.
Rug is じゅうたん (Jutan)
Armchair is ひじ掛けいす (Hijikake isu)