I am trying to solve the problem of very long response times from MySQL when opening a connection using the MySQL Connector for .net.
I have installed MySQL 5.5 running
I can explain your question "the length time it takes to get the connection to MySQL Open after no calls to the database have been made for about 3 or 4 minutes. If no database calls are made for a few minutes it takes 8 or 9 seconds or more to open the connection again." why it happens:
The Windows Azure websites uses concept of hot (active) and cold (inactive) sites in which if a websites has no active connection the site goes it cold state means the host IIS process exits. When a new connection is made to that websites it takes a few seconds to get the site ready and working. While you have MySQL backend associated to this website, it take a few more seconds longer to get the requested served as there is some time taken by IIS host process to get started. That is the reason after few minutes of in activity the the response time is longer.
You can see the following presentation for more details on Windows Azure Hot (active) and Cold (inactive) Websites: http://video.ch9.ms/teched/2012/na/AZR305.pptx
As this time, I am not sure and do not know how you can keep the websites always hot, even if moving to shared website or it is not possible at all. What I can suggest you to write your issue to Windows Azure WebSites Forum and someone from that team will provide you an appropriate answer.