Learn Icelandic: Þau eiga að ryksuga gólfið. - They are supposed to vacuum the floor.

You are right that eiga by itself means "to own" or "to have." But when you combine it with að and another verb in the infinitive (like að ryksuga), it changes meaning. The pattern eiga að means "to be supposed to" or "ought to" do something. It expresses an obligation or expectation.

Why is the pronoun þau used for "they" instead of þeir or þær?

Þau is the neuter plural pronoun for "they." In Icelandic, if a group of people includes at least one male and one female (a mixed-gender group), you always use the neuter þau. You would only use the masculine þeir for a group of just men, or the feminine þær for a group of just women.

The word ryksuga looks interesting. What does it literally mean?

It literally means "dust-sucker"! It is a compound word made from ryk (dust) and suga (to suck). It acts as both the noun for "vacuum cleaner" and, like in this sentence, the verb for "to vacuum."

What case is gólfið in this sentence?

Gólfið is the definite form of gólf (meaning the floor). It is in the accusative case here because it is the direct object receiving the action of the verb ryksuga (vacuuming). Since gólf is a neuter word, its nominative and accusative forms look exactly the same!