Yes, that is not a problem whatsoever. I have a huge experience working with children and know how to get their attention and make the lessons super fun. We take breaks, speak, and play mini-games related to chess that improve the child's logic and critical thinking.
Though it is true that depending on the age, sometimes the child finds it very hard to focus, but for that there is a simple solution too. We can simply make the lessons 30-minutes long, so the child doesn't lose interest. 
ABSOLUTELY NO! That's what chess used to be 30 years ago, children thought that the game was meant for adults, and it wasn't that popular in between them.
THE TIMES HAVE CHANGED. Chess has become one of the most famous sports/games in the whole world, and is now seen as a great method of making friends, getting smarter, more patient, etc.
If possible, I recommend to have at least 2 lessons a week to see significant and very fast progress. This will all be talked through and planned with the parents, considering school, family activities, clubs and other stuff.