Well, my point was that your aggregation formula
SUM(video_quartile25_rate*impression)/SUM(impression)
is essentially correct, but beware of the field impression. Make sure you ONLY include impressions for YOUTUBE_WATCH, and NEVER include impressions of the other networks.
What you could try is use the Web UI, select the fields like you have in your opening post. Then add segmentation for Network. Add a filter for one campaign.
Try to see if you can figure out the summary results for the quartile - shouldn't be hard because there's only one campaign.
Next, change the filter, add a second campaign. See if you can still determine the summary results. Then a third. Try a campaign that contains not only youtube network, but also other networks.
Remember: segmenting is the important thing to do here. Without segmenting you probably cannot validate the aggregation formula.