How to adjust salaries of more than two countries to a common year given that the data was collected in different years?
Your three stage procedure is not correct. 

In effect, since the data is already in USD, to apply PPP you need to revert it back into LCU (local currency units). That is your step 1. Have in mind this might introduce errors if the one you use is not the same the authors used. 

Then, since wages are not for the same year, ideally you do need a proxy for wage growth of each occupation, for each country and year, so that you can compute wages in LCU for the same year. You are attempting to do this in step 3 using GDP growth, which is not ideal (and the order is wrong too). I would rather use an inflation index. This is not ideal either, but the World Bank provides an homogenised one. Alternatively, you could get wage growth for occupational groups (or aggregate) from national statistical offices, but this is a time-consuming exercise.

After putting wages into the same year, you proceed with step 2, where you convert LCU to USD PPP, using a common year.