Is the ordering of Brodmann areas arbitrary? A single Brodmann area is defined based off cellular composition. Are the Brodmann areas ordinal, and if so for what reason? (e.g. is there something that makes Brodmann area 1 the "first one", and area 2 the "second one"?)
The place to find this answer is of course Brodmann's original 1909 manuscript Vergleichende Lokalisationslehre der Großhirnrinde: in ihren Prinzipien dargestellt auf Grund des Zellenbaues  available here in German  or here in English (Brodmann's Localisation in the Cerebral Cortex: The Principles of Comparative Localisation in the Cerebral Cortex Based on Cytoarchitectonics). 

The English version is unfortunately not open access (and I don't have the inclination to pay US$90 for it) and the German version is a scanned copy of the original printing, so there's no easy way to search the text without reading all of it. However, skimming through it, I see that the description of the areas (which he calls Felder, lit. 'fields') begins in Kapitel IV (Chapter 4), page 130. He mentions that this work is largely an extension of his previous 1907 work, but beyond that there doesn't appear to be any mention of why he choose his numbering, at least not that I could find with a quick skim. It appears though that he choose a more or less arbitrary starting point and worked more or less counter clockwise from there, with a few small exceptions.

Elsewhere in the manuscript, Brodmann discusses the division of the cortex into larger homologous regions in both humans and other mammals (p. 129), and it's clear from the description of the areas that they are not divided up based on any study of homology. For example, he mentions that the regio postcentralis (lat. "post central region") is divided up into the cytoarchitectural areas 1, 2, 3 and 43 (pp. 130--132). The regions appear to correspond loosely with our modern notion of "lobes", but tend to be somewhat smaller (for example, there are additional subdivisions near the central sulcus).