Fondant recipe with egg white. Are there health risks?
Powdered sugar is about 3% corn starch by weight.  So the candies set due to a combination of three factors:


Drying
Protein network from the egg white albumen 
Thickening provided by the corn starch


There is some minimal health risk from the raw egg, as it is possible it was infected with salmonella.  The actual incidence, at least in the US, is very low, but you will have to assess your own risk tolerance.

Given that these confections are close to pure sugar, they should keep close to indefinitely, as long as they are not gotten wet--but they will be very hydrophyllic, and would absorb water even from humid air.  So keep them in a sealed dry container.