Assuming you have done everything correctly inside Radiant...
 
Once finished in radiant press save and exit. (It will ask you about exporting lighting if you have made any changes to lighting just click OK and it will exit...)
 
Then in the Mod Tools you will need to select your...

I would recommend starting out by watching "IceGrenade" on youtube. He helped me get going as a mapper by watching his tutorials from the very beginning. The questions you have asked are hard to answer without writing a page essay.
 
Zombie spawners, barriers etc are very simple ...