The prevalence of ubiquitous computing enables new opportunities for lung health monitoring and assessment. In the past few years, there have been extensive studies on cough detection using passively sensed audio signals. However, the generalizability of a cough detection model when applied to external datasets, especially in real-world implementation, is questionable and not explored adequately. Beyond detecting coughs, researchers have looked into how cough sounds can be used in assessing lung health. However, due to the challenges in collecting both cough sounds and lung health condition ground truth, previous studies have been hindered by the limited datasets. In this paper, we propose Listen2Cough to address these gaps. We first build an end-to-end deep learning architecture using public cough sound datasets to detect coughs within raw audio recordings. We employ a pre-trained MobileNet and integrate a number of augmentation techniques to improve the generalizability of our model. Without additional fine-tuning, our model is able to achieve an F1 score of 0.948 when tested against a new clean dataset, and 0.884 on another in-the-wild noisy dataset, leading to an advantage of 5.8% and 8.4% on average over the best baseline model, respectively. Then, to mitigate the issue of limited lung health data, we propose to transform the cough detection task to lung health assessment tasks so that the rich cough data can be leveraged. Our hypothesis is that these tasks extract and utilize similar effective representation from cough sounds. We embed the cough detection model into a multi-instance learning framework with the attention mechanism and further tune the model for lung health assessment tasks. Our final model achieves an F1-score of 0.912 on healthy v.s. unhealthy, 0.870 on obstructive v.s. non-obstructive, and 0.813 on COPD v.s. asthma classification, outperforming the baseline by 10.7%, 6.3%, and 3.7%, respectively. Moreover, the weight value in the attention layer can be used to identify important coughs highly correlated with lung health, which can potentially provide interpretability for expert diagnosis in the future.