Are the limits of your yard shown in the picture or does your yard extend into what looks light it might be a wooded area on the left? It would be good if the picture showed 15' farther in all directions.

A typical lawn surface does not provide shock absorption safety provisions.
Yes, grass is definitely better than concrete but still falls short in the safety division when compared to playground fiber ( play grnd fiber is usually cedar chips that have been put through a tumbler to eliminate splinters ) It is also recommended that the fiber be laid down at an appropriate depth of no less than 9 inches. ( There are government guidelines for both residential and commercial playgrounds to help both parents and designers _ http://www.cpsc.gov//PageFiles/122149/325.pdf )
As far as the cushioned space around the play structure I believe a mimimum of 8 feet in all directions is commonly recommended.

We design a lot of playground structures both on the commercial and residential levels and I have to say that an organic shaped surface area that is totally integrated into a dynamic and creative space is much more visually and functually stimulating than just a square plop of playground fiber surrounding a play structure.

If you have a local library or bookstore nearby I highly recommend Taunton Press's books, " Kidspace Idea Book" and " Outdoor Living Idea Book". There are lots of creative outdoor playground spaced in the books that will inspire you to think beyond the square within the box.