You can use factor scores as depended variable for CHAID analysis.
I don't understand what is the factor value mean. Factor scores (regression, bartlett or other methods) or classical sum scores can be used. For more clarification, you can use sum of items which is belong to same factor.
assuming you are using factors as predictors and you have a categorical outcome, you sure can. the interpretation of your split points on factors are not trivial though. You can look at the factor loads for that specific factors to see which specific original variables determines that factor. This would provide some level of understanding. However, there may bean infinite number of combination of your original variables that would make your factor to take the value of 0.282. So, the interpretation depends on the nature of your original data and not easy.