No, heat has no influence on the strength of a magnetic field produced by current flowing around a coil of wire. The strength of that magnetic field is strictly the result of the ampere-turns of current going around.
However, heat can effect the magnetic permeability of various materials. If the electromagnet has anything other than a air core, then how the field resulting from the ampere-turns is concentrated and shaped can differ with temperature. This concentration and channeling of the magnetic field can make a electromagnet appear to have a stronger field, and can make it act "stronger" in many applications.
For example, let's say you wrap 100 turns of wire around a wooden rod and put 1 A thru it. The magnetic field strength is strictly a function of the 100 ampere-turns of current going around. However, this magnet will be able to pick up heavier objects if the wooden rod is replaced by a iron rod of the same shape and size. This is because the iron is a much better conductor of magnetism than air and wood are, so the magnetic field lines will be concentrated at the ends of the iron rod. This more concentrated field is able to pick up heavier magnetic objects as a result of this concentration, even though the overall magnetic field has the same average strength in both cases.
In the example above, the apparent strength of the electromagnet with a iron core depends on material properties of the iron, which can vary with temperature. The magnetic permeability of free space is not effected by temperature, so the same coil without a core would make a magnet that does not vary with temperature.
Of course extreme tempertures change the wire and will eventually melt it so that you don't have a electromagnet anymore at all. That obviously changes things, but I'm assuming that's not the kind of effect you are asking about.
If you have an H-bridge controller that can switch cleanly at a PWM frequency which is sufficiently fast relative to the motor's inductance (the lower the inductance, the faster the PWM must be), driving it with a waveform that's 60% forward and 40% reverse will be a good way to drive it forward at 20% speed; 40% forward 60% reverse will be a good way to drive it backward at 20% speed. If both conditions above are met, driving a motor in this fashion will give a speed response which is much more linear than PWM'ing between driven and "open-circuit", and will also be more energy-efficient. Additionally, trying to drive the motor at a speed which is somewhat slower than it's presently turning will provide regenerative braking [i.e. allow motor energy to be fed reasonably nicely into the supply].
The important thing to note is that running the PWM too fast for the H-bridge controller may waste energy in the H-bridge controller; running it too slow for the motor inductance will increase the amount of energy wasted in the motor. If the PWM is much too slow, driving the motor at half speed may use many times more energy than trying to run it at full speed. If, however, the motor is driven with a fast PWM and the H-bridge can handle it, efficiency may be very good; a stalled motor driven at 75% forward 25% reverse will have about half the torque as would one driven at 100% forward, but will only take about a quarter of the power [about 75% of the time, it will draw about half as much current from the supply as it would if on 100%, and the other 25% of the time it return that same amount of current].
Best Answer
This is no different than controlling a brushless DC motor open loop. Actually, this is a brushless DC motor.
I would not use a single electromagnet. With a single pair of poles, you can only make the field flip, not actually spin at some controlled rate. Or put another way, you can make the field rotate at any frequency you want, but it won't rotate smoothly. It will always jump 180° at a time and the torque goes to 0 twice per cycle.
The minimum to make a rotating field requires 3 poles, which is why so many motors have 3 wires going to the field windings. The three windings would be arranged in a even "Y" layout. You can connect the inside ends of each of the coils together and control everything from the 3 remaining outside connections. Each of these 3 lines is then connected to a half bridge, meaning a high side and a low side switch that can be controlled independently.
Since the rotation speed is going to be very slow compared to the speed of a microcontroller, you have the micro step thru all the phases of a rotation sequentially. The easiest is a 12 step process. Each line is driven with a OPEN - HIGH - OPEN - LOW repeating sequence. Draw it out on paper and you can see how you make one change to one line each step, which eventually walks the magnetic field thru steps of about 30° each. One advantage of this 12 step scheme is that it naturally does break before make. There are other scheme that can deliver more power thru the same set of windings, but the 12 step scheme is very easy to implement and more forgiving of screwups, making it all around better for beginners.