Although weather being bad, we decided to go hiking.

本句为错误。改正如下：

Although weather was bad, we decided to go hiking.

Weathing being bad, we still decided to go hiking.

Mike went homeward alone, while his heart full of strange emotion.

本句为错误。改正如下：

Mike went homeward alone, while his heart was full of strange emotion.

Mike went homeward alone, his heart full of strange emotion.

独立主格结构前不可加任何从属连词。这是原则问题。