Spiders have mouths. :)

They either use it to drink directly, or they get enough water from their prey that they don't have to drink, or they consume parts of their web to get water, or they shovel water into their mouths with pincers.

I like spiders.

Swedish does not have the distinction between drinks and is drinking that English has, so you cannot directly translate it to the latter, there is only spindeln dricker.

Vattnet.

Well, the English construction isn't "a noun performing a verb", it's is plus a participle. In English, they use is plus the present participle to express continuous aspect, so that they can distinguish between he is drinking and he drinks. In Swedish, we don't have that distinction, we just have the present tense dricker which covers both.

It's an ett word and it works like this:
ett vatten 'a water'
vattnet 'the water'
vatten 'waters'
vattnen 'the waters'