Solo los domingos para público en general. Si eres maestro la entrada es gratuita todos los días.

Taxis are fine as long as you agree on a price beforehand plus they will wait to take you back to your resort just so they get the fare. Much less stressful unless you are used to the bus system.

El Rey is a fine local attraction, but is really nothing like as impressive as Chichen Itza. It's much closer, much cheaper, not crowded, much, much smaller and has less impressive ruins. I can't speak for the museum as I didn't visit it.

Coming from the airport, it's to the left (opposite the ocean) about a kilometer or so before you start to see hotels. I walked there from the Royal Islander resort. It's near Playa Delfines and the Cancun block letters that people take pictures in front of. There is good signage for it, so you will see where it is when you drive into the hotel zone from the airport. Hope that's helpful!

$50

There is some parking right in front of the gate.