What is the expiation for a false oath made out of ignorance, and is it permissible to pray an obligatory prayer during a prohibited time?

There is no expiation for your oath, because it falls under the category of a 'vain oath' (laghw al-yameen), which is when a person swears about something they believe to be true, but it turns out to be otherwise. This is the opinion of the majority of scholars. However, if you expiate as a precautionary measure, it is good. If you swore that you heard the news from a specific source, then there is no expiation due upon you. It is advised to avoid narrating everything one hears. As for making up missed prayers, it is permissible at any time of the night or day, even during the disliked times (awqat al-nahy), and this is the opinion of the majority of scholars.