Neither carbon steel nor cast iron should ever be washed in a dishwasher. For more tips, check out our guide on how to season a cast iron skillet. If cost is a significant factor in your decision ...
About 5 minutes at 325 F should be plenty. After it's dry and while it's still hot, you can (and should) feel free to season it more with a cast-iron seasoning wax (I like Made In's mix of beeswax ...