Yep! By the office/visitor center.

Yes, spigot next to one of the vault toilets.

I've only dry camped there among the house-sized rocks, but Google says that there is water. "The campground has drinking water, hot showers, flush and vault toilets. Group camping is also available. City of Rocks State Park also has a nice visitor center, day-use/picnic area, campfire center, botanical garden, interpretive trails/tours, 5.5 miles of hiking and mountain biking trails. City of Rocks State Park - Campsite Photos, Reservations

Yes

There is drinking water. See: https://www.emnrd.nm.gov/spd/find-a-park/city-of-rocks-state-park/

Yes, you can.