The Stabilizer does wonders but cannot perform miracles; in other words not every clip is suited to be stabilized. If its too wobbly or whatever it does not work.
For transitions to work properly you need to trim them first.
Adobe Premiere Pro Help | Transition overview: applying transitions
In your export you are going from HD widestreen to SD 4;3. Hence the poor quanlity.
What is the end file for: Youtube? If so use a Youtube preset from the H.264 format.