Learn Portuguese (Portugal): Ele está a despejar o café no lava-loiça sujo. - He is pouring the coffee in the dirty kitchen sink.

You would say: Ele nem sequer está a despejar o café no lava-loiça sujo. When nem sequer (not even) is placed before the verb phrase, it acts as a negative word on its own, so you do not need to add não to the sentence.

Does despejar just mean 'to pour'?

It means 'to pour' in the specific sense of emptying, dumping, or disposing of something. Because he is pouring the coffee down a dirty sink, despejar perfectly captures that he is getting rid of it. If he were pouring coffee into a mug to drink, you would use servir or deitar instead.

Lava-loiça is a compound noun. How do I make it plural if there are multiple sinks?

The plural is lava-loiças. When a compound word in Portuguese is made of a verb (lava, from lavar) and a noun (loiça), only the noun takes the plural 's'. Also, note that loiça is the standard European Portuguese spelling, whereas louça is more commonly seen in Brazil.

Could I use the preposition para instead of em (no) here?

Yes, you could say para o lava-loiça sujo, which emphasizes the direction or destination of the action (into/towards the sink). However, using no (the contraction of em + o) is perfectly natural and very common in Portuguese to indicate the container or location where the liquid ends up.