How to extract/cut out parts of images classified by the model?
Your thinking is correct, you can have multiple pipelines based on the number of classes.

Training:
Main model will be an object detection and localization model like Faster RCNN, YOLO, SSD etc trained to classify at a high level like cat and dog. This pipeline provides you bounding box details (left, bottom, right, top) along with the labels.
Sub models will be multiple models trained on a lover level. For example a model that is trained to classify breed. This can be done by using models like vgg, resnet, inception etc. You can utilize transfer learning here.
Inference: Pass the image through Main model, crop out the detection objects using bounding box details (left, bottom, right, top) and based on the label information, feed it appropriate sub model and extract the results.
I am new to deep learning, I was wondering if there is a way to extract parts of images containing the different label and then feed those parts to different model for further processing?
For example,consider the dog vs cat classification.
Suppose the image contains both cat and dog.
We successfully classify that the image contains both, but how can we classify the breed of the dog and cat present?
The approach I thought of was,extracting/cutting out the parts of the image containing dog and cat.And then feed those parts to the respective dog breed classification model and cat breed classification model separately.
But I have no clue on how to do this.