Learn Russian: Сундук стоит напротив кровати.

The preposition напротив (opposite / across from) always requires the noun following it to be in the Genitive case. Since кровать is a feminine noun that ends in a soft sign (-ь), its Genitive ending changes to -и.

Why do we use the verb стоит (stands) for a chest? Wouldn't it just 'be' there?

In Russian, pieces of furniture are usually described using verbs of physical position like стоять (to stand), лежать (to lie), or висеть (to hang), rather than just using a verb for 'to be'. Because a chest (сундук) has a solid, flat base or short legs that it rests on vertically, it 'stands' (стоит).

Does напротив only mean 'opposite' in a physical sense, or can it mean 'opposite' like an opposite idea?

In this sentence (and as a preposition of position generally), напротив is strictly used for physical location (across from / facing). If you want to say 'the opposite of' for abstract ideas, you would use a different word, like the noun противоположность.

Could we put напротив after the noun instead, like кровати напротив?