Can airplanes transmit/relay flight data to ATC's periodically while in flight?
No, there's no data relayed like this to ATC. All ATC really gets is the filed flight plan, and then the transponder data. ATC just cares about organizing all the traffic in the skies, and making sure it is safe, orderly, and expeditious.

Some airlines use a system similar to this to relay maintenance data to the airline from the aircraft, that's how AF447 was found to have something wrong, and how they generally found it's location quickly(I think within 48 hours debris was found from the a/c due to the location info through ACARS(Aircraft Communication and Reporting System)). This system, and it's capabilities are optional and variable depending on what the airline wants to spend. This is why there's so much controversy over the loss of MH370, since they didn't have the full reporting schedule, and a very basic version.