Is "during her reign" same as "while she was reigning"? Can I also say "over her reign" (like "over lunch")?
Yes, "during her reign" is the same as "while she was reigning" here. "During" has the same meaning as "while". You can also say "over her reign" here since "over" in this context also expresses the meaning of during a period of time. Ex: During my time in college, I traveled to many different places. Ex: He visited France over the summer.