I am quite happy with all the progress you guys have made in forming an idea of this Empire.
I agree with both das and NK on many points, though on the question of religion, I am inclined to agree with das. The largest and most stable Empires of our history have always been inclined towards a limited secularity, with religion being used as a tool of the state. However, over time, the Empire will manage to overcome all of the challanges that have ever faced it, and shall rule supreme. This would, in the eyes of Empire, probably require some justification, and inevitably, minds would run towards the superiority of the people of the Empire, and thusly, a superiority of their leaders and government, perhaps even creating the belief of a divine force which has allowed the people of the empire their greatness. This belief would likely be militant in nature, similar to, as das said, the OTL cult of Mithras. However, in order for there to be a strong enough anti-military and secularist movement, the religion cannot be too supreme over the citizens of the Empire, as there would not be any significant anti-military movements in such a case (as the other elements of society would simply view the revolutionaries as insane heretics). The religion must be, IMHO, quite ingrained into the psyche of the common man (and widely believed as fact by the lower classes) yet at the same time, the higher elements of society (the aristocracy, or the commercial and military elite) shall remain widely secular, whilst still promoting the religion to further their own success and the success of the empire. Thusly, the anti-military movement must be restricted to the upper echelons of society, until a great change utterly disrupts the society of the Empire, wiping away the foundations of the past and causing wide levels of discontent and chaos throughout the lower classes (for Japan, it was the defeat in WWII, for the Empire, it will likely be internal - as it has achieved such a complete hegemony over the world).
I do not really see the need for a matriarchal society, as the larger empires of OTL have been forged by patriarchal societies. Women are inclined, naturally, towards the more emotional. Men, towards the more practical and mathematical. If we require the formation of a large, overbearing empire, a patriarchal society shall suit us best, though in said society, woman should be granted considerable rights and oppurtunities for success. (as in Rome)
As for the development of tanks - I do believe they will be created, though I do not believe they will be widely used by the empire, or if they are, will not be heavy in nature, and will not be the main force by which the empire relies upon its power. The Empire will, as I see it, rely on its immense forces of sea and air to bring about the total destruction of their enemy's capability for war, and thus, bring about a forced surrender. The majority of the land units shall be mobile assault forces, deployed from the skies and the coasts in order to capture and destroy strategic locations throughout an enemy's territory and infrastructure (as such, these forces would remain relatively concentrated in nature, though not small, as the empire commands such huge manpower reserves and must be on guard accross the whole of the world). On the Northern masses of land however, I do see a future for tanks and larger vehicles, though these would be specialized to these areas. There would be no need for nukes, as the empire has achieved such a level of control where these are not neccessary, and that nukes, if ever spread through the world, would bring about a statemate, and thus keep the power of the empire in a limited check, as it could no longer stop upon any state it choses.
They should not, and will not, IMHO, be developed any time soon - until perhaps the empire reaches a state of total desperation, and begins a range of odd projects (similar to the later years of Nazi Germany).
In the navy, I do think Carriers (or some form of them) would be quite useful, but due to the limited prestige factor, I do not think they will be the flagships, nor shall they get the most emphasis. The empire, through its vast, and seemingly unexhaustable resources, will invest in things which communicate its prestige on a much grander level, and thusly, I see the creation of a core fleet of floating fortresses, which coordinate and operate as the center of the Empire's military operations. The floating fortresses would act as a mobile bases to control the navy, and the smaller fleets would be made up of the assault craft (such as missle cruisers) and the support craft (such as aircraft carriers). The primary emphasis of the navy would be to bring about an enemy's destruction on land and on the coast, as the Empire will, by now, achieved near complete control of the seas, excepting perhaps the pirates.