1. Say: "All that stuff is mine." ("Stuff" as a noun is plural.)
2. Sometimes either could be used. (But in the previous sentence we wouldn't say might instead of could.) "It could happen" means there is a possibility that the thing will happen, and in that sentence we could use "might" instead of "could" with "might" indicating a greater likelihood that "it" would happen.
3. "She wouldn't believe" is more indicative of a refusal to believe than "She didn't believe", but the result is the same. :)
4. Use "in a while" to indicate the time that has passed between events and "for a while" to indicate the duration of an event.