Archiving data in synthetic DNA offers unprecedented storage density and longevity. Handling and storage introduce errors and biases into DNA-based storage systems, necessitating the use of Error Correction Coding (ECC) which comes at the cost of added redundancy. However, insufficient data on these errors and biases, as well as a lack of modeling tools, limit data-driven ECC development and experimental design. In this study, we present a comprehensive characterisation of the error sources and biases present in the most common DNA data storage workflows, including commercial DNA synthesis, PCR, decay by accelerated aging, and sequencing-by-synthesis. Using the data from 40 sequencing experiments, we build a digital twin of the DNA data storage process, capable of simulating state-of-the-art workflows and reproducing their experimental results. We showcase the digital twin's ability to replace experiments and rationalize the design of redundancy in two case studies, highlighting opportunities for tangible cost savings and data-driven ECC development.