In my personal experience with Bentley, the check doesnt do anything to keep them in a straight line. He still keeps his head turned to the side sometimes, he is getting better though now that i know how to hold his head properly with the reins. But, no, the check doesnt do that.
I might be wrong, but i think they can buck with an overcheck, depending on what length you have it. Im training Jazzy to drive and I had the check too loose (fixed it) and he started thrashing his head, but my friend was helping me lead while i was in a cart so if he did she would know how to stop him, considering she has horses herself. And they can most definatly rear LOL! I have seen videos of that happening, its not a very great thing to see....