Should I keep the microwave turntable and plastic turntable support ring in the microwave oven when using convection mode?
I cannot vouch for your specific GE oven, but I've been using my combination oven with the turntable and turntable support in Microwave, Convection Bake, Convection Roast and Combination Fast Bake for the last >10 years with no ill effects.

As your oven has the same technical capabilities as mine:


  Combination Fast Cooking
  Your oven also offers the option of Combination Fast Cook, using microwave energy along with convection cooking. You cook with speed and accuracy, while browning and crisping to perfection. 


I'm led to believe you're safe to use the turntable in convection only mode like I do as my round shelf fits perfectly on the glass turntable, so I never asked myself the question and have just used it like that since the first time I turned on the oven.