Hi Kristen! I activated international calling on my cell phone so I didn't use the room phone or a calling card so can't help you out there.
The best place I ate in Playa was Carboncitas. The prices were great and the food was fabulous! It is at 5th Avenue and 4th Street. As you are walking down 5th Avenue when you reach 4th Street, just look down the street and you will see it just a little ways down. We are returning to Playa in July with the rest of the family and I am definitely taking my husband and son there!
Yes, there is sometimes a lot of traffic to cross and I can understand your concern with a 6 year old. I never felt in danger, but you may have to just check it out for yourself.
I considered it to be way too far to walk to the front entrance. We always took a shuttle from the Mayan Palace lobby to the front entrance.
Yes, the rooms have coffee makers - thank goodness!

They provide a package of filters, a couple of packets of creamer and sugar, and one day's worth of coffee. You can buy more coffee in the convenience store, but it is expensive. So I just took coffee in my suitcase.
No, they don't have DVD players. We took our portable DVD player and those red/yellow/white cords to attach it to the TV.
My biggest tip (you may have already read this on Trip Advisor) is to get get a room in buildings 1-7. Until I got there, I had no idea just how important this would be to me! We were in building 2, which worked out perfect. Buildings 1 and 2 are connected and share an elevator and concierge in the center. I walked thru building 1 and literally was at the Grand Mayan pool. Just a short, beautiful walk around the corner and you are at the Mayan Palace Pool, the beach, the convenience store, the spa, and the restaurants. It was great! The first 7 buildings form a "C", with 1 and 7 on the ends. But there is a lagoon in the middle of the "C" so you can't just cut across. Building 7 wouldn't be bad because you would just walk around that end of the lagoon. Then you have buildings 8 thru 13 strung out lengthwise behind the "C". The temporary Grand Mayan lobby is past building 13. So when we were headed to the lobby on the shuttle, we would see all these families at the shuttle stops with their hands full of beach stuff. I have no idea how prompt the shuttles were to those buildings, but hopefully they were quick! When we wanted to take a shuttle to the Grand Mayan lobby or the front gate of the resort, we went to the Mayan Palace lobby and caught it there. They came along every few minutes and we never had to wait. If you don't get buildings 1-7, don't be sad! It is still a beautiful resort with great service and the shuttles were very efficient!
Another thing we did that would be handy with your child is to take a couple of insulated cups. I would fill mine half full with water and then fill it up with ice at the ice machine on my way out of the building. Then I would have ice water to take to the pool. Later during the 11am-1pm happy hour (2 for 1 drinks!) I would pour out any remaining water and fill the cup with my frozen drinks. When you buying during 2 for 1, they bring you both the drinks at the same time so my 2nd drink would melt before I finished my 1st one.
Hope this helps! Let me know if you have any more questions. I am so happy to help someone since I got so much help myself from this forum!