Deep learning (DL) models for disease classification or segmentation from medical images are increasingly trained using transfer learning (TL) from unrelated natural world images. However, shortcomings and utility of TL for specialized tasks in the medical imaging domain remain unknown and are based on assumptions that increasing training data will improve performance. We report detailed comparisons, rigorous statistical analysis and comparisons of widely used DL architecture for binary segmentation after TL with ImageNet initialization (TII-models) with supervised learning with only medical images(LMI-models) of macroscopic optical skin cancer, microscopic prostate core biopsy and Computed Tomography (CT) DICOM images. Through visual inspection of TII and LMI model outputs and their Grad-CAM counterparts, our results identify several counter intuitive scenarios where automated segmentation of one tumor by both models or the use of individual segmentation output masks in various combinations from individual models leads to 10% increase in performance. We also report sophisticated ensemble DL strategies for achieving clinical grade medical image segmentation and model explanations under low data regimes. For example; estimating performance, explanations and replicability of LMI and TII models described by us can be used for situations in which sparsity promotes better learning. A free GitHub repository of TII and LMI models, code and more than 10,000 medical images and their Grad-CAM output from this study can be used as starting points for advanced computational medicine and DL research for biomedical discovery and applications.