I'd say since all humans and animals work the very same way with some minor variations, the brain seems pretty much deterministic.
Yes and no.
The brownian motion involved in chemical signaling is random, which affects neuronal behavior. However, this system of unreliable parts gives rise to an overall reliable machine. So, brains aren't strictly deterministic in terms of their basic functioning, but they are deterministic enough for us to actually understand what it's doing.
Incidentally, computers function in a similar way at the physical level. A transistor doesn't know what 0 or 1 are. It's simply a switch that flips to a different position when it has a high enough voltage. High voltage is 1, low voltage is 0. Since voltages are variable thanks to noisy power supplies and other factors, the threshold between 0 and 1 has to be carefully set, otherwise the CPU won't work. Once again: a reliable machine generated from unreliable parts (and energy inputs.)
That's not to draw too direct a comparison, as human brains are essentially sophisticated analog signal processors and computers are digital number crunchers, but it's interesting that they both emerge from similar principles.
All input is processed the very same way in every normally functioning human brain.
Not exactly. The devil is in the details. Ten people, all with "normal" brains, will not respond the same way to having their fingertips poked with a needle at the same pressure. Some will react sooner than others. If you introduce external noise (such as vibrating the person's hand), that may improve or reduce their ability to sense the needle. (I believe some research says it actually improves threshold detection.)
knows anything about stochastic resonance, maybe she can comment more on this.