Learn Russian: Твоя сумка под тумбочкой? — Нет. - Is your bag under the nightstand?

Because of the preposition под (under). When we talk about the location of something (where it is), под requires the Instrumental case. For feminine nouns ending in -а like тумбочка, the Instrumental ending is -ой.

Will под always be followed by the Instrumental case?

Not always! It takes the Instrumental case when answering the question 'where?' (location), like in this sentence. If you are talking about motion toward a place (answering 'where to?'), like the action of putting a bag under the nightstand, под would take the Accusative case instead.

Why is there no Russian word for 'is' in this sentence?

In Russian, the present tense of the verb 'to be' (am, is, are) is almost always left out. So instead of saying 'Is your bag under the nightstand?', you just literally say 'Your bag under the nightstand?'.

The word order here is exactly the same as a statement. How do we know it is a question?

In Russian, you do not need to change the word order or add helper verbs like 'is' or 'do' to make a basic yes/no question. In writing, the question mark tells you it is a question. In speech, you just raise the pitch of your voice on the key word you are asking about.