Can lying flat on my back also add pressure to my diaphragm and give me trouble breathing?

Overweight. Yes mostly, if you are overweight or having heart failure. Please realize sleeping on the back (like babies autonousmly do) facilitates sleep in insomniacs.
Yes. Yes especially in the second and third trimester. Try propping yourself up with some pillows. Also eat small meals and make sure several hours have passed since your last meal and when you go to bed.