Is it possible to record the order participants answer questions?

This may not get you all the way there, but you could use some javascript to record timestamps for when each question was answered:
https://community.qualtrics.com/discussion/3612/recording-timestamp-for-item-responses-on-a-single-page
You'd just need an additional step of data processing after that if you want a clean "1, 2, 3" ordering for each case. Maybe others would have ideas for how to do this within QTX? Otherwise I'd do that step in "post" (e.g., using the RANK formula in Excel).