No it would not. Referring to any clock on board your ship would reveal no changes. You would conclude that time was 'flowing' in a perfectly normal manner--one second per second. Only by reference to an external clock could you 'perceive' a change in the "rate of flow" (whatever that is) of time. Furthermore, you would reasonably conclude that it was the external clock that was speeding up or slowing down, since manifestly things are normal aboard ship.
"Besides, time does "flow" normally, as a dropped ball shows the direct flow of time (the ball doesn't hover or reverse directions)..."
The laws of physics are time-symmetric. You cannot distinguish a film of a ball bouncing up from the pavement from one of the ball dropping to the pavement and played backward. Entropy provides an 'arrow' of time but this is not an explanation; it is an observation. It is perfectly possible, e.g., for a glass of water at 70F to spontaneously freeze, or all the air in a room to collect in one corner. The second law is empirical, a statement that we rarely see such phenomena--indeed that they are so improbable as to require more than the age of the Universe to be likely to happen.
Exactly. Just like in your example: only by referencing the ground outside of the train could you see space "flow" by. Space-time is a four coordinate system. Our only problem is that we can't displace back in time, nor control our personal perception of time. Just like we are always "here," but need an outside referent in order to make "here" a meaningful statement...
Yes. But causality is always a one-way street, flowing from cause to consequence. I have no support on this in any of the physics literature, nor do any of the truly knowledgeable people on this board agree with me, but I regard causality as the principle "arrow" of time, and the only one which gives time what we perceive as its direction -- from past to future.
I believe you are incorrect. The laws of macrophysics (mechanics) are time-symmetric. Several quantum mechanical operations have recently been proposed as non-time symmetric (such as K-meson decay). At this point, I am reaching the edge of my expertise (the last time I took a physics course... double-majored physics/math for a while... was several years before the experiments quoted), but several experiments at CERN (see this) and elsewhere have suggested that certain weak force interactions violate charge and parity invariances, which means they must have an equal-sized time variance to balance out the CP violations, in order to remain CPT invariant.
Either way, both CPT invariance and time invariance tend to be premises in order to hypothesize further, as opposed to confirmed fundamental laws. But universal time invariance is most certainly not accepted to the point of saying that "the laws of physics are time symmetric" ...