These are great questions. I think these questions are centered around one core question: how do we reconcile the statement that the speed of light is a constant, and the fact that the speed changes when we change the reference frame.

Allow me use some simple equation:
our daily life experience indicates that, if a person is moving with speed u relative to earth, and a train (a reference frame) is moving with speed v relative to earth, then the person is moving at:
speed u' = u - v relative to the train.

Or in other words, if the person is moving at speed u' relative to the train, the person is moving at speed u = u' + v relative to earth.

This simple transformation is called the Galilean transformation, it looks so natural and consistent with our daily life experience, so we never question its validity. However, quite counter-intuitively, the Galilean transformation is only valid when u, u', v are far smaller than the speed of light c. When one of them is close to c, the Galilean transformation will be replaced by a new set of transformation called the Lorentz transformation.

Lorentz transformation is bit complicated, so I will not write down the equation, but under the Lorentz transformation, indeed no matter how we change the reference frame, the speed will never exceed c.

For example, if v = 0.5 c, u' = 0.5 c, (both speeds are half of the speed of light), the speedu (person relative to earth) should be u = 0.8c according to the Lorentz transformation. But if we use the (incorrect) Galilean transformation, u would be c.

If v = 0.6 c, u' = 0.6 c, the speed u should be u = 0.88c according to the Lorentz transformation. But if we use the (incorrect) Galilean transformation, u would be larger than c.

It is very hard to accelerate objects in our daily life to near speed of light, so it is hard to test the Lorentz transformation. But for tiny particles this is feasible, and the super colliders do this every day. The validity of the Lorentz transformation has been tested and confirmed with these fundamental particles.