Solubility and Evaporation - Would the concentration still be 1 ppm? A certain salt has a solubility of 1 mg/1 L.
I have 1 mg of this certain salt dissolving in 1 L of water, I have a concentration of 1 ppm. 

1st Question: 
If the solution is left open to evaporation, 0.5 L of water (50 %) evaporated, would I still be getting concentration of 1 ppm while 0.5 mg of this salt became solid again and left at the bottom of the solution? 

2nd Question: (if it's a yes to the 1st question)
If I then fill the container back to 1 L mark with water, would the concentration be 1 ppm again after the salts are completely dissolved?
Correct, when the water evaporates you'd have more salt than is soluble in water. The excess would then fall out of solution. Adding the same amount of water that evaporated would leave you with the same kind of solution you started out with, so the concentration of dissolved salt would be the same as when you started (although you might need to stir some to get the salt back in solution).