Shakes

Vibrations that only occur at higher speeds are usually due to a wheel out of balance. A broken tire belt can cause that but you'll usually feel that at low speeds too. An inspection at a tire and alignment shop should identify the cause.

Absolutely not. An anti-sway bay doesn't do anything when the car is going straight. During hard cornering when the body wants to lean, an anti-sway bar tries to hold the two corners of the car at the same height. They have a designed-in amount of flex, so they only reduce the amount of "body roll"; they don't totally eliminate it.

Nothing about an anti-sway bar relates to road speed or vibration. Any shaking or vibration has to involve parts that are rotating. That includes tires with broken belts, brake drums and rotors that have chunks missing due to rust, or welded-on weights that fell off, or to a lesser extent, half shafts. Half shafts rotate at the same speed as wheels and tires, but because their diameter is so small, they rarely cause a vibration unless they're bent, and I've never seen a bent half shaft except on badly-crashed cars.