Can neural networks be adapted without recreating them completely?
The answer to your question is "Transfer Learning". 

Since the datasets "cat and dog" and "mouse" are quite similar as both are images. In DeepNet for recognising "cats and dogs", any deep learning network in its early layers learn to identify low-level features like edges, etc. It learns high-level feature like eyes, ears, etc in few further layers and in very later layers of DeepNet, it starts to recognise the intended object. In DeepNet for recognising "dogs", similar pattern will follow. On comparing these two, one may find that first few layers of DeepNet in both cases produce similar low-level features like edges, etc. 

Hence, first few layers of the DeepNet learned from "cats" can be used as base layers for the DeepNet for "mouse" detection. This technique is called transfer learning. 

To apply transfer learning, dataset on which DeepNet has been trained on and dataset on which this technique can be applied must be similar. 

This transfer learning video by Andrew Ng would also be helpful in understanding the concept