Is a girl forbidden to a man if he swears, "If you do such and such, you are forbidden to me until the Day of Judgment," and it becomes clear that she did only a small part of what he swore upon?

Prohibiting a non-mahram (definition) woman is null and void, and nothing results from it, unless one intends ẓihār by it, in which case it becomes binding according to some scholars. Then, the expiation for ẓihār becomes obligatory before touching her if you marry her. Performing the expiation is more cautious. What matters in knowing whether an oath is broken is the intended meaning of the statement. A Muslim should choose a righteous wife and avoid words of prohibition and divorce.