Solar Eclipses for dummies: Step 1 - moon equatorial orbit around a planet without axial tilt

1. Every new moon would be a total solar eclipse.

Incorrect. Even if the Earth had no axial tilt and the Moon had an equatorial orbit, that doesn't mean that every new moon would be a total solar eclipse. Yes, the moon would always generate some kind of an eclipse at every new moon, but it's orbit shape hasn't changed -- it's still an ellipse. It's further at some points in its orbit and closer at others. When it's further away at new moon, it appears smaller than the sun in the sky, making an annular (ring) eclipse, not a total eclipse.

2. The path of totality would always cover the same area: a 250km wide corridor with its centre at the equator (so 125km over the northern hemisphere and 125km over the southern hemisphere).

For the same reason as (1), the path of totality would vary from a maximum amount down to nothing. For annular eclipses there would be a similar path of annularity instead. But whichever path it is would always be centered on the equator.

3. I think the path of totality wouldn't always cover the same area, e.g. always over Africa, so Q1: Is there a simple formula to predict which areas would be in the path of totality?

I'm sure that there are formulas that would predict which areas would be in the path of totality/annularity, but they would not be simple. They would depend on exactly when new moon occurs and the exact distance of the moon from the earth. That would control which strip along the equator would see the eclipse and how wide it would be. Because the moon slows down in its orbit when it's further away from the earth, and it speeds up in its orbit when it's closer to the earth, the time between new moons is not constant. The fact that Earth's own orbit around the sun is also elliptical, and it also slows down/speeds up when it is far/close from/to the Sun complicates this as well.

4. In real life, the moment of totality can vary from seconds to seven minutes. But in my study example, Q2: the period of totality should always be the same, right? Because the crossing path of moon and sun is always the same (while in real life the Moon can overlap the Sun at different angles and going in different directions) Q3: How can I calculate this period of time?

Again, because of the elliptical orbit of the Moon, the length of totality or annularity still can vary. Even during a single eclipse, the length of totality depends on your exact location on Earth. Your exact location determines your exact distance from the moon during the eclipse, which will vary depending on whether the Moon is getting closer/farther and how fast. There are formulas for calculating this, but I'm sure they are quite complicated even with the moon orbiting over the equator and with Earth having no axial tilt.

5. What about the latitudes for whom the solar eclipse would only be seen as partial? I understand that the nearest to the path of totality, the more covered the sun will be, and vice-versa. Q4: Is there a set value that says e.g. for each km, another degree is visible?

Generally, the closer you are to the equator, the more the sun would be covered. But this would vary slightly depending on the distance of the Moon from the Earth, and it wouldn't depend solely on your latitude. As Gerald points out, the moon's shadow, in which you would see a total solar eclipse, is cone-shaped. Different points on the Earth close to the equator would experience a different cross-section of the cone, which may be narrower or wider depending on your exact location. Where the eclipse is partial, the apparent size of the moon would control how much of the Sun is covered. A further-away moon would appear smaller in the sky and thus cover less of the Sun. Close to the borderline where the Moon appears to graze the Sun, the Moon may miss the Sun entirely if it's far enough and it appears small enough.