Does tribulation expiate all of a believer's sins and transgressions, including prayers he did not perform or fasting he did not make up, or does it expiate some of them but not others?

Calamities atone for minor sins. As for major sins, such as intentionally abandoning prayer, the majority of scholars hold that they require specific repentance, along with making up missed prayers and fasting the days one broke their fast or delayed making up.