I don't think it's provable in this form, you'd also have to specify what "single algorithm" is. Because algorithms themselves are delineated by human categorizing; what you call a "single" algorithm can be subdivided into several simple(r) algorithms, and at the same time can be just one part of what someone else will call "a single algorithm".
Not necessarily. We got a mix of innate head-start and training from parents/others for a really, really, long time before we could do that. The brains learning style and speed even changes over time going from rapid learning in intuitive way to learning slower and with more stability in a combo of self- and others-driven way. At some point, our mind is pretty well-formed where our own thoughts and explorations are driving much of our behavior. And somewhere between the kid and adult parts we start easily learning new skills.
There's probably a mix of general, specialized, and stuff in between algorithms. It changes over time, too, instead of being a static, pre-trained model.
You could also easily have different algorithms for different senses, evolved at various historical stages...