Osteochondrodysplasia, affecting 2-3% of newborns globally, is a group of bone and cartilage disorders that often result in head malformations, contributing to childhood morbidity and reduced quality of life. Current research on this disease using mouse models faces challenges since it involves accurately segmenting the developing cartilage in 3D micro-CT images of embryonic mice. Tackling this segmentation task with deep learning (DL) methods is laborious due to the big burden of manual image annotation, expensive due to the high acquisition costs of 3D micro-CT images, and difficult due to embryonic cartilage's complex and rapidly changing shapes. While DL approaches have been proposed to automate cartilage segmentation, most such models have limited accuracy and generalizability, especially across data from different embryonic age groups. To address these limitations, we propose novel DL methods that can be adopted by any DL architectures -- including CNNs, Transformers, or hybrid models -- which effectively leverage age and spatial information to enhance model performance. Specifically, we propose two new mechanisms, one conditioned on discrete age categories and the other on continuous image crop locations, to enable an accurate representation of cartilage shape changes across ages and local shape details throughout the cranial region. Extensive experiments on multi-age cartilage segmentation datasets show significant and consistent performance improvements when integrating our conditional modules into popular DL segmentation architectures. On average, we achieve a 1.7% Dice score increase with minimal computational overhead and a 7.5% improvement on unseen data. These results highlight the potential of our approach for developing robust, universal models capable of handling diverse datasets with limited annotated data, a key challenge in DL-based medical image analysis.