My *very limited* understand of SDE is that it is just the data server between ArcGIS desktop and a relational database that lets the software "talk" to the relational database. ArcGIS for Server is the software that is installed on your physical server (or an Amazon server if you go that route) that lets you manage web services, mobile apps, enterprise geodatabases, and server-side processing. Depending on how your organization is set up, you may need to get whoever manages your server to help you out with publishing rights.
For AGOL, if you want to link up to features or services that will reside on your server (AGOL users see and edit the data, but the data is actually stored on the server) you will need to get the rest services link out of your ArcGIS Server Service Directory and stick it here: [ATTACH=CONFIG]27094[/ATTACH]
If you want your feature stored on AGOL (not on your server) you can batch run "Package Layer" and "Share Package" from Data Management Tools.tbx to quickly get a bunch of layers out there.