Implementing an Avalanche Photo Diode Circuit
Is this 10% error an additional error to the amplification error that results from temperature differences?

Yes. The IC mirrors the APD current, and you measure this second current. The given error applies to this current.

If you look at figure "APD Current Monitor Accuracy vs Temperature", you'll notice that 10% is an extreme value for a temperature of +125°C and an APD current of 250nA. If you stay below +100°C, the error seems to be less than 0.5%.

In addition, the error is more or less constant for a given current, (and a reasonable temperature range for your project), so your measurement value will always differ from the ideal value by a constant factor. 

I think this error is negligible compared to problems arising e.g. from temperature dependency of the APD itself.

The LT3482 is only capable of a 90V output. Is this too low?

Probably yes...

When a Photon hits the APD, it creates one free electron (and a hole), which move to the terminals of the APD. On its way, the electron passes a region of high electrostatic field, where it gains the energy to create an avalanche of secondary electrons. Finally, the gain of an APD is the number of electrons generated from a single incident photon.

From the APD datasheet, the APD is fully depleted at about 80V. (See the capacitance vs. voltage plot. When it's depleted, the capacitance stops decreasing.) At this point, the APD already works like a photo diode, (i.e. gain is about 1). To get a gain > 1, you need to increase the voltage. As you said, the datasheet suggests a voltage above 130V. Extrapolating the gain curves of your last figure, the gain will be about 1 at 90V, you you will have a photo diode.

Now, it still depends on the amount of incident light. If it is large, your very low gain may be OK, but if it's small, you definitively would like to have some gain. What's large, and what's small is determined by the sensor itself and your mechanical setup, and of course your probes.

Finally, an APD used as photo diode is expensive and complex (high voltage, temperature dependency, ...). In this case, you'd better get a standard photo diode.
If you need the gain, you will need a higher voltage. May be, you also have a look at other APD manufacturers, as far as I know, Hamamtsu uses relatively high voltages.