Yes, 梦 can be a verb and a noun. For instance, like you said 我梦见/到了 我的小狗. Here 梦见/到 is a verb phrase. It's strange if you only leave 梦 here to be a verb. We don't say 我做梦我的小狗, but we say 我做梦梦到了我的小狗(The first 梦is a noun, while the second one is a verb). As a noun, 梦 can be said in this way: 我做了关于我的小狗的梦. Mostly, when we want to decribe what we dream about, we will use the first way "我梦到了*/ 我梦见了………".
Maybe I've made this a little difficult for you to understand? Anyway,hope this will help ya. Please feel free to tell me if you still don't make it.