How many partitions can a Table have?
For a given account, what are the maximum number of partitions, and tables that can exist?
Can any of these limits be increased, if so, to what number?
There's no defined limit on the number of Partitions you can have per table or the number of tables per storage account. It is limited by your storage account size (100 TB). You may find this blog post useful: http://blogs.msdn.com/b/windowsazurestorage/archive/2010/05/10/windows-azure-storage-abstractions-and-their-scalability-targets.aspx.