Why is the deprotonated form of dopamine more reactive than the protonated/neutral form of dopamine?
In general the free amine will react as a nucleophile i.e. by "donating" an available lone pair of electrons. In its protonated form, this lone pair is "blocked" and thus it has to lose a "blocking" proton to react (I use the words in "quotes" more freely - although I'm a chemist and not biochemist but hopefully it will help).

However re oxidation of dopamine, the reactive site is not the nitrogen (at least not the primary one - but will come back later) but the catechol part. Because of the electron donating effect of the 2 neighboring OH's, the aromatic ring is electron rich and more prone to oxidation to give o-quinone products. This oxidation reaction depends to an extend to the pH but mainly to the presence of (obviously enough) reactive oxygen species (ROS) and also enzymes (oxidases) or even Fe ions that can catalyse this oxidation. 

Coming back to the nitrogen, after oxidation occurs with creation of reactive sites on the quinone part, it can act now as a nucleophile and attack, creating indolines that will further oxidize to the more stable but toxic indole products.