Do the compressors and turbine rotate at the same speed?
That's one badly-worded quote:


  Single spool jet engine. These are those jet engines who have one shaft inside them. The compressors and turbines rotate on a same shaft. 


Correct. In a single-spool engine, all of the compressor and turbine stages rotate at the same speed because they're attached rigidly to the shaft. 


  The speed of compressors and turbines decreases half way down the compressor stages, that is why they can’t match the local airflow. 


No, the speed of the compressors and turbines doesn't change. It's the airflow that changes, and causes a mismatch.