If a space ship accelerated constantly, would its astronauts constantly feel the forward movement?

Suppose the spaceship was accelerating constantly at 1g, what would that feel like? Well Einstein gave us the answer to that: it would feel exactly like standing on the surface of the Earth where the acceleration due to gravity is 1g. This is (one statement of) Einstein's equivalence principle. If the acceleration were continued for many hours that wouldn't be any different to being on the Earth's surface for many hours, which of course we all do.

I'll try to give as simple an answer as possible to each of your three questions.

1. The astronauts will always feel the acceleration (but as Muphrid pointed out, in this case it's tiny and might not even be noticed)
2. It is not possible to feel speed while in a spacecraft. Astronauts in orbit travel at 28000 km/h but feel absolutely nothing, even if they're outside. Similarly, inside a car you do not feel the speed, only the change in speed (i.e. acceleration - and note that acceleration can be in any direction: forward, backward, left, right, up or down). You feel the speed only if traveling through the air, where you feel the air dragging at you.
3. Speed does not cause any harm at all, as you never feel any of its effects.

Now a couple of provisos.

First, if your speed became a large fraction of the speed of light, any particles (even single atoms or nuclear particles) you encounter would fly clean through the spacecraft and through the astronaut. If you're going fast enough, these particles in effect become nuclear radiation, and that can indeed harm you.

Also, it is not possible, even in theory, to keep on increasing your speed by 100 km/h every 10 minutes. You will never be able to go faster than 300,000 km/s even if you keep on accelerating forever. The same force will give you smaller and smaller acceleration as you go faster. This is what Einstein discovered with his theory of Special Relativity.

You're describing constant acceleration, and this would result in the astronauts feeling a constant force in the direction of travel exerted on them by the spacecraft.

This particular acceleration is actually quite small--about $.005g$--and as a result, it might hardly be noticed, let alone be dangerous.