Learn Dutch: Steken de bijen de imker vaak? - Do the bees sting the beekeeper often?

Look at the verb! steken is the plural form. Since de bijen (the bees) is plural and de imker (the beekeeper) is singular, de bijen must be the subject doing the action. If the beekeeper were doing the stinging, the verb would be singular: Steekt de imker de bijen?

Why is the adverb vaak placed at the very end of the sentence?

In Dutch, adverbs of time and frequency like vaak (often) usually come after a specific direct object. Because de imker (the beekeeper) is a specific object, vaak is pushed to the end of the sentence.

I thought steken meant to put or to stick. Does it also mean to sting?

Yes, it means both! For insects or sharp objects, steken means to sting or to stab. But it is also heavily used in everyday Dutch to mean to put or to stick something somewhere, like je sleutels in je zak steken (putting your keys in your pocket).

Is the noun bijen (bees) related to the common preposition bij (at/by/with)?

They just happen to look exactly the same. The singular noun is de bij (the bee), which becomes de bijen in the plural. It has no grammatical connection to the preposition bij.