I do understand that the main problem right now is the fact that you don't get new mail after the mail application first opens. But you also had problems with 2 other mail applications, so I don't know how they all relate. This problem could be caused by many different things including some email providers that don't work well with apple, settings you are using, etc.
Here is something basic to try. Quit all applications and open disk utility. Select your boot volume and click on Repair Permissions. Once done, restart the computer and open Mail. See if it works.
If not, create a new user account for testing and log into that account. Create the mail account in that user account. Test to see if it works there. This will tell you if there is a problem with the user account or not.
Post back with the results so someone can go from there.