"The dog and the cat are sleeping together in a basket."
Translation:De hond en de kat slapen samen in een mandje.
Dutch has a continuous but it is formed "zijn aan het [verb]". But it is not necessary to use the continuous here because the Dutch present simple can translate to both present simple and present continuous in English depending on the context.
"Katje" is the diminutive of "kat". The diminutive versions, words ending in -je, take the article "het". Therefore it should be "het katje".