If you have a 100W electrical load and you drive 100W plus efficiency losses, say 110W, into the generator, things will be in a state of equilibrium, with 100W being converted from mechanical input power into electricity, and the other 10W of mechanical input power being eaten up by losses.
Now suddenly put 1kW of mechanical power into the machine; at that instant, before the rotational speed can change, the 100W electrical load will continue to present the same mechanical load to the prime mover. Things will not be in equilibrium, and the machine's rotational speed will accelerate. Depending on circumstances, this may or may not increase the electrical load. Certainly the generated voltage will go up, and any simple resistive load will therefor absorb more power, but maybe you have some regulation such that the load continues to draw exactly 100W.
So assume the load continues to draw exactly 100W. Where does the extra 900W of mechanical power go then? The machine's speed must increase until the losses equal the mechanical driving power; so it ends up turning extremely fast, the increased power going into increases in friction in the bearings, windage loss due to the rotating parts, eddy currents in the magnetics (and doubtless a couple of other things I forget at the moment), none of which are desirable.
You would find that, without exceeding the machine's electrical rating, you would quickly exceed its mechanical ratings, i.e., probably long before you got to 1000W, the rotation speed would be several times the suggested speed, and catastrophic failure would likely result. Note you can do this with no electrical load on the generator at all.
Windings that are side by side that carry current in opposite directions cancel each others magnetic field - you need to wind the electromagnet so that current travels in the same direction in all turns of the coil. There is no sensible argument for doing differently.
The strength of the magnetic field is related to number of turns and ampere's flowing. For the same number of turns wound on a longer solenoid the field reduces with solenoid length.
These are your only options other than to use superconductors! Fewer turns can be compensated by greater current or reduced solenoid length (stacking turns).
Here's a calculator you can use to estimate field strength and it uses a field value called permeability and this means you can check on various materials and choose one that gives the highest strength BUT for household coil making a piece of iron is fine.
You can also enter the diameter of the coil.
Best Answer
Firstly, current in not induced; voltage is induced. You get current because of the induced voltage in the receive coil is connected to a load (or a short circuit in some examples).
The ferromagnetic core attracts lines of flux from the magnet and concentrates that flux through the receive coil (more so than if the coil were air-cored). That means you get a bigger induced voltage when the magnetic flux is changing.
This usually means more current produced when you load the coil.
However, because the inductance of the receive coil has increased (due to the presense of a ferromagnetic core) and that component of inductance is in series with the induced voltage, the current produced into a load may not be any bigger when the magnetic field changes are very fast.