Learn Russian: Они часто водят нас осматривать этот храм и статуи. - They often take us to examine this temple and the statues.

The key is the word часто (often). It tells us this is a repeated, habitual action, not just a single trip happening right now in one direction. For repeated trips, we use the multidirectional verb водить (here conjugated as водят for "they"), rather than the unidirectional verb вести.

Does водят нас mean they are driving us there in a car?

No. When taking a person or animal somewhere, водить specifically means to lead or take them on foot. If they were driving us in a vehicle, the sentence would use возят (from возить, to transport by vehicle). Note: водить can mean "to drive" only when the direct object is the vehicle itself, like водить машину (to drive a car).

Why is there no preposition before осматривать? How does it mean "take us TO examine"?

In Russian, you don't need a preposition to express purpose after a verb of motion. You can simply put the infinitive verb directly into the sentence. So, водят нас осматривать perfectly translates to "take us to examine" or "take us in order to examine."

What case is этот храм и статуи and why didn't the endings change?