I'm aware you can get triggers from other sources and that in general the trigger syncronizes the acquisition with a trigger event. But as for its purpose, is it always to sync the acquisition with the input?
The answer is no. A trigger can be related to one input. But it can also be related to other triggering sources, line another input of the scope, or an external triggering signal, etc.
Sometimes. For some measurements you want to trigger on the level and slope of a signal that you're trying to see on a scope. At other times, you may want to take a look at a signal when a particular combination of digital signals occur. If your scope is equipped with digital inputs, this is relatively easy to do.
Of course, another type of triggering is external triggering. The purpose of external triggering is to synchronize an oscilloscope measurement with another instrument. For example, you might connect the trigger output of a signal generator to the external trigger input of an oscilloscope so that the signal generator can tell the scope when it has begun outputting a test signal.