If you are flying a plane at 350mph does a tail wind with a velocity less than your groundspeed or airspeed assist you, and if so how?

If you are flying at 350 mph that's through the air. If the air itself is moving at 50 mph then your actual movement on the ground would cover 400 mph. It does not matter if the tail wind is less than your ground speed.

And if you do a 180 degree turn and moving 350 mph into the headwind, your groundspeed will be 300 mph.

And if it hits you from the side you get this Airplanes flying sideways - YouTube