First and foremost, slow down the ball machine a bit so its not spitting out one ball after the other. Work on quality, not quantity. If you wanna set it up where the ball you hit crosses the baseline on the other side and the ball machine spits out another so its like a rally then start moving your feet more. If you wanna work on technique then slow down the interval a bit.
Ok, now first off the thing that I would work on with your forehand if you took a lesson would be to get your feet more wider apart. Your feet are rather close together, at all times. This is one of the causes why you don't and can't utilize your lower body much into your stroke as you could be. You hit some semi-open stance forehands as well but even then the feet should be wider. Wider during the ready position, keep them wide when you move to the ball and a nice wide base as you hit the ball.
Right now from the forehands in the video that I saw you're all arm and it happens because your feet are too close together. You won't be able to really use your core and quads efficiently unless you widen that stance. Tennis is played with your legs and core. You generate power and control from the ground up and that to me is your biggest flaw on your forehand.
Another clue that you do not use your lower body in the stroke is how you sorta lean over to hit your stroke. You gotta line up that right leg with the ball, and push off of it. What you're doing is sorta falling/leaning over to your right as you hit your shot. There is no loading in your legs, or rather as much as there should be. You generate your power from your arm, which is wrong. You can't really load because again your feet are too close together.
Regarding your shoulder turn as mentioned earlier, I don't think its that bad. The angle of the camera makes it look like its less than it really is. However, like I said before everything starts from the ground up and because of your feet being too close together, you cannot get a fuller shoulder turn as it will put you off balance. You in turn compensate this by generating power just with your arm, and your shoulders doing minimal work, your legs almost nothing except a bit of a weight transfer thats it. You mentioned hitting too flat, and this is the reason why.
Watch Hass here how wide his feet are, lower center of gravity and how he uses his legs to start the stroke
http://www.youtube.com/watch?v=9NY0ZJbVMTY
Here's Nick B. explaining from the ground up
http://youtu.be/cbS0ycWJJEw?t=6m16s