First a few lines of basics. If you put a loop into the magnetic field and this loop turns within it, the magnetic flux through loop shall change according to the formula
$$\Phi_B = \vec{B} \cdot \vec{A} = B A \cos\phi = B A \cos\omega t,$$
where $\vec{B}$ is magnetic field strength, $\vec{A}$ is area of the loop and $\phi$ is angle between $\vec{A}$ (perpendicular to loop) and $\vec{B}$, while $\omega$ is angular velocity of the rotation of the loop.
If you use coil with $N$ loops, then induced voltage on the coil shall be
$$\mathcal{E} = N \frac{\text{d}\Phi_B}{\text{d}t} = - N B A \omega \sin\omega t.$$
Therefore, yes, if you use twice larger magnetic field, you get twice larger voltage.
However, twice larger magnetic field does not necessarily mean it is twice as hard to turn. For example, if you have no electric load on the generator, there is practically no current in the coil and there is practically no Lorentz force! However, if you have completely ohmic load, voltage twice larger means current twice larger and yes it becomes twice as hard to turn.
In practice, you should also consider the internal friction of the generator. So even if there is no load, some muscular power will be required in order to overcome friction. When you increase the electrical load, required power in order to turn generator increases.
Of course, it is difficult to keep rotation constant, which means that with larger angular velocity $\omega$ voltage increases. To keep generator's output voltage constant, you need some electronic circuit, of which the simplest possible includes zener diode.
Best Answer
The electromagnet appears to contain a ferromagnetic core. This core is magnetized by the electromagnetic coil. It retains a magnetic field after the coil is turned off, and its field may take some time to reverse after the coil polarity is reversed.
Moreover, if the test magnet is more powerful than the coil (which sounds to be the case), it will cancel the field of the coil completely. You are essentially attaching the ferromagnet to the permanent magnet.
Source: Wikipedia.
You might try again with only a coreless coil. Perhaps try an inductor rather than a dedicated solenoid.