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In Azure there is a difference between a Storage Account and a container.
A Storage Account is a collection of containers - theoretically only limited by the storage limit within an Azure account.
In your question you do not appear to be creating the container "mycontainer" within your Storage Account. This would explain the "The specified container does not exist" exception - until you explicitly create a container, it does not exist.
You can check that your container exists from within the Azure Portal - within a Storage Account you will see a link to containers - under that link is a list of all the containers for that Storage Account.
If the container is missing, you need to create it - from the tutorial you linked, this line of code needs to be executed at least once:
$blobRestProxy->createContainer("mycontainer", $createContainerOptions);