According to my understanding...
You can install all this software on Amazon machine.
Regarding connection-- You can insert ArcGIS Server WMS to QGIS (Using ADD WMS Layer)
You can not publish data from QGIS to ArcGIS Server. Its totally different architecture (open source/closed Source)
Regarding PostGIS Yes..you can install it on AMI Machine and you can connect from ArcGIS desktop to PostGIS (Add Spatial Server option in ArcCatalog)
If you want to publish data then you need to put your data in ArcMap than publish as map service (to ArcGIS Server & then create the Web application) but from Q-GIS you can not do these tasks..
I hope this will help you..let us know if you have any questions..thanks