my team uses the Youtube API for a 2 week contest, where participants have to send a video. We want these videos to be hosted on Youtube, only for 3 or 4 weeks.
However,
How long does YouTube have to respond to this form and what are the delay for increasing the quota?
It really depends upon the API. Some can take as long as two months others take a few weeks. So expect it will take a while.
We have a billing account, what is the price of a quota increase over 3 or 4 weeks if it's necessary to pay ?
To my knowledge the YouTube API is a free API. To my knowledge there is no public pricing for it.
However if they contact you can work with a developer relations manager to increase your API quota for a price. It isn’t documented because it varies based on an application-by-application basis.
Do you think it is possible to contact live (phone / chat) to speed up our request?
Since you have requested a quota increase. Developer relations may contact you and work with you on giving you what you need.