Col. MacKenzie first stores the frozen sperm in the JAG kitchen refrigerator and then in a home freezer. Neither would be cold enough to preserve the sperm at liquid nitrogen temperature.
When Harm visits Palmer in prison, the warden assures Harm that Palmer has no connection to the outside on the computer. However, Palmer claims to have downloaded the op-ed from the newspaper (as was confirmed by Agent Nichols). That would only have been possible if he had a connection to the outside to get to the newspaper website.