Alfred didn’t give Dick the money cause he was wise with money, but because he thought Dick was genuinely good and so he can trust that what would be done with the money would be genuinely good. The whole dynamic with the idea is that Dick isn’t someone from money. But a kid from a circus who was uncomfortable seeing someone do his dishes.
Also with the kids, like you said Dick was shocked. He didn’t know what do in the moment. As he was shocked. It was only after that he could think on it and ask what he can do for them.