No, the battery will not be damaged by the low voltages. It will only charge when the voltage is above the batteries; the higher, the more current. Any voltage below that will cause a negative differential, which would normally cause a negative current except the diode is preventing that. The circuit is pretty much in a "stalled" state at that point and does nothing.
Well, without a DC/DC converter, what you'd get is below a certain speed=no load, then at a certain speed, with a strong low-resistance generator, you'd hit a "brick wall" where more work gets more resistance but no more speed.
Which isn't necessarily a bad thing because humans need to pedal at about 90 rpm for trained cyclists, and 60-70 rpm for non-bikers to get optimum leg output. Faster wastes leg energy and less means that the leg will have to pump against too stiff of a resistance load to generate the same power (half the rpm requires twice the torque to generate the same power). Maybe at cool-down, a person should have a reduced load at lower speeds, but at "power", it needs to be ~90 rpm.
A DC/DC can change the generator output to whatever the batt needs. Also, how does a car alternator work? It's always ~13.6v over a huge rpm range! Well, that doesn't have a fixed permanent magnet making the field. It has a separately excited field coil- an electromagnet. The voltage produced is proportional to the magnetic field strength, so it can adjust voltage through that field winding. It takes a number of amps IIRC to run the field winding, but the driver's not a linear reg, it's a DC/DC chopper. The field winding spends much of its time with its terminals shorted together which, in an inductor, causes the current to circulate and only decays as per the I^2*R losses in the winding. As such it doesn't actually require that much power to run.
In fact a car alternator, with a regulator, might be a good starting point for you, although that falls short of the "90 rpm regulation" requirement.