Learn German: Die Toilette ist neben dem Bad. - The toilet is next to the bathroom.

The word neben (next to) is a two-way preposition. When describing a fixed location where nothing is moving to a new place (answering where? or wo?), these prepositions require the dative case. Since das Bad is a neuter noun, it changes to dem Bad in the dative case.

Can I use die Toilette and das Bad interchangeably like we say bathroom in English?

Not usually. In English, we often say bathroom even if we just mean the toilet. In Germany, homes often have the toilet in a separate small room from the shower or bathtub. So die Toilette (the toilet) can literally be next to das Bad (the room with the bath).

Would it ever be correct to say neben das Bad?

Yes, but only if you are describing movement towards a destination. If you were carrying a portable toilet and putting it down next to the bathroom, you would answer the question where to? (wohin?). Then you would use the accusative case: neben das Bad. Since a normal toilet is permanently installed, we use the dative for location instead.

Is Bad short for another word here?

Yes, das Bad is the common short form of das Badezimmer (the bathroom). Both are neuter nouns, so you could also say neben dem Badezimmer and the grammar would stay exactly the same.