Joe there are plenty of free video programs. Just make sure they have slow motion and frame advance. YouTube has slow motion and it is easy to upload especially if you want to share.
As to how to analyze corners. If it is simply the shape you want the judge's position, but if you want to really see what the plane is doing film from upwind just far enough back to get the whole maneuver and at 90 degrees to the wind with the plane coming at you. If you are using the judge's position, make the bottom of the frame the pilot's feet. Most people don't do the second two so let me tell you what it shows me. The downwind when run in slow motion gives you how the plane is coming out of corners much better than the upwind. It also helps if you have a windsock in view. In slow motion you can see control deflection and skid. The 90 degree is my favorite in diagnosing corners. You get to see the plane entering the corners and every bit of roll and yaw shows but best of all you see the line rake from the side and what that nasty wrist is up to. You don't get any of that with the traditional position. Nothing can replace slow motion for measuring skid. You either trim it out or learn how to present it like it wasn't there.
Another thing you should do every now and then - a zoom in on you from downwind. You will see things you swear you weren't doing!

I use a product called VLC. It has slow motion, frame advance and you are able to repeat any segment in a continuous loop which really helps.
I am only addressing the video just make sure you are getting enough to evaluate the plane, the positioning and the pilot.
Good luck - Ken
Oh, one more thing, you don't need service on a smart phone to use the camera. I have two and one of them is just for pictures.