In regards to radiation exposure and the Van Allen belts for the Apollo missions; the missions were relatively short (up to 10 days for the moon landings as compared to years for a mars mission), they took trajectories that would avoid the most intense exposure, they accepted risks that would not be acceptable today, and to a large extent they only found out what the radiation environment was like up there by going.
Here is a pretty good read that I'm not even going to attempt to summarise (apologies to the no click rule):
http://www.braeunig.us/apollo/VABraddose.htm
For Lunar dust - again, the missions were relatively short, and equipment was designed to withstand a certain amount of wear, but even so you can see heavy wear and dirt on the suits and gloves used in the missions. Wear that is acceptable during a 10-day mission becomes a big problem for longer missions.