LaurieB

I have 14 and 11 yo boys.  Like others have said khaki pants and polos in the MDR.  On formal night they wear khaki pants and a button down shirt.  Immediately after dinner they change back into their athletic gear to go to the teen/kid club.

The "smart casual" style for regular nights in the MDR and specialty restaurants definitely doesn't mandate jackets or ties. I'm planning on wearing chinos and a polo, or chinos and a long-sleeve button down for non-formal in the MDR and any specialty restaurants we visit, you can bet your kids will be just fine with khakis and a polo.
For reference, there's a pretty helpful video from Cruise Tips TV on YouTube, titled "What to Wear on a Royal Caribbean Cruise". Gives visual examples of different outfit types that include some young boys / young men's outfits.

Laurie, you and i would get along great (says Jane wearing her mickey earrings and bracelet).  Im an uber planner who loves disney and just came back from an 8 day spring break trip to the world.   I would kill to do a disney cruise, but there is no way to afford it like i can royal.  I went on my first cruise 2 years ago on royal and have loved it!  The mdr and windjammer have great food in my opinion,  but you will find that food is subjective.  We have really enjoyed trying the specialty restaurants, but you can totally survive without them.   I haven't been on the big ships (freedom is the largest I've been on) and my understanding is that they definitely need more planning for dining and shows.  However,  my guess is you can relax just like on any other ship and they have a ton for the kids.  My 14 year old has yet to be bored, even on the smaller ships,  and she absolutely loved the teen center.   The kids area are amazing on the royal ships.    I can't do any comparing to disney since i haven't been lucky enough to go on them,  but if you have any specific questions ask away here....lots of us have had amazing experiences on royal and can advise away!   Have a great trip!   Jane

Absolutely not. Just like on DCL, the specialty restaurants on RC augment the fabulous complimentary dining.
Nearly all the former DCL folks that try RC remark something along the lines of, "the food was WAY better on RC than on DCL".  You can very easily enjoy just complimentary food on your RC cruise have a a satisfying experience. I love the main dining room, and the Windjammer buffet offers a very wide assortment of choices. 
Generally, I recommend trying a specialty restaurant or two to spice things up a bit.  It's not absolutely necessary, but I think it helps.  Think of it like going to Disney World and sprinkling in a table service restaurant among a trip filled mostly with counter service restaurants.  It's just a nice way to make a night or two extra special.

Absolutely not!!  IMO, the MDR food is fantastic, and if I get something I don't care for, I can just get something to replace it.
 
Laurie - leave your alter ego at home and just relax.  As soon as I step onboard, I can literally feel the stress leave my body and mind.
You and your family will have a great time!