I am attempting to use MS Project Professional 2013 to send out automated outlook e-mails the week before a task is due among other automated e-mail tasks. Is this possible with the desktop program or will I need something like MS Project Server or the MS Project web app to achieve such a task?
You can set up these functions using VBA but the problem you will run into is that the file has to be open for it to check the conditions and run the code to send an email. Sharepoint would actually be the right service for setting up those types of automated tasks.
What you might consider doing is assigning tasks to each member of your team via outlook. If you don't mind everyone seeing all of the tasks, you can create a group calendar in Outlook and put all of your project tasks in it.