What's the best strategy to train a CNN with images that only have labels for positive characteristics?
I think your best bet would be transfer learning. Start with a model that has already been trained with a wider dataset such as the ones presented here. From there you can train the model with your specific dataset. You can then use output nodes for the labels which you have available to you, and you can get the predictions for the other images from the pre-trained model which are usually trained for thousands of different classes. 

Alternatively, you can train a model with all the output classes you have in your label set and another output for "other". Then when an output node is selected you can pass that same input to the pre-trained model which was trained with your data as well.