1. I would use "thrives" and adjust the phrasing slightly
"She showed that she thrives on challenging situations"
or if speaking of a specific situation:
"She showed that she thrived on the challenging situation"
2. "can thrive" suggests the potential to thrive, but what we want to say is that she does thrive.
"chips were down" is past tense, but we're speaking in the present tense.
"We realized that she thrives when the chips are down."