I dreamed that my brother and I were running for our lives, trying to understand our dreams.

This shows that you have some pressure and want to escape, but you can't be so rational. Your brother appears in your dream, which shows that your brother also has this kind of pressure, which can be judged as the pressure of study. Running to grandma means you want to go back to your childhood, and grandma won't take you in, which means you subconsciously know there's nowhere to run. That dog symbolizes your life now-just like some movies use rags to represent farmers. Don't worry too much, it's no use running away. If you escape today, you can't escape tomorrow-if you don't master scientific and cultural knowledge today, you may be more sad in the future.