It seems like \'select count(*) from c\' in the SQL queries allowed by documentdb in the azure site and through the documentdb explorer (https://studiodocumentdb.codeplex.com/)
My code count solution is also working...once I just selected the id as a Papa Ours pointed out :) To get my original post to work, replace this line:
var documentCount = dc.CreateDocumentQuery(update.SelfLink, "SELECT * FROM c").ToList();
with this line:
var documentCount = dc.CreateDocumentQuery(update.SelfLink, "SELECT id FROM c").ToList()
I still like the idea of the stored procedure as it will work in the documentdb studio (really cool project :)) - https://studiodocumentdb.codeplex.com/