Is it likely that, at the culmination of the Sol War, both sides will believe that they have won?
At the moment of the UEF's formal surrender, both the UEF Council of Elders and the Security Council will believe they are on track to a victorious resolution, although their criteria are way different.
1: How will the heat death of the universe affect the Shivans?
There's not likely to be a heat death in current cosmology; the Big Rip should come first. The Shivans wouldn't be badly affected by the universe coalescing into a bunch of black holes, although their manifestations would change wildly.
This next ones a bit out there. 
2: Let's say I (out of spite) encrypted "Alpha 1 giving middle finger.png" into the DNA of a highly extremophilic Terran microbe artificially modified to live in the atmosphere of a Y-class brown dwarf; and thus allowing them to survive into heath death longer then any other form of life; would these "Terran lifeforms" survive the Shivans via technicality and be able to mock/insult the Shivans on a personal level via Alpha 1's angry likeness living on in their genetic code?
Or is this quest for galactic bragging rights just a hollow victory gone to far? 
As long as they're not a threat to cognitive diversity they should be just fine.
How's Bosch doin'?
Bosch said he knew humanity's future lay with the Shivans, that a tragic era of hatred and misunderstanding was coming to an end, and (later) that Laporte had to destroy the GTVA. His encounter with the Shivans was obviously violent, but Bosch and several others were taken away to form a component of an apatic anima. If Bosch genuinely believed in Neo-Terra (a dubious proposition), could the destruction of the GTVA somehow serve that goal? Could Bosch still usher in the end of an era of hatred and misunderstanding, presumably to be replaced by the absence of hatred and the presence of understanding? Understanding of what?\
What were the Vishnans trying to accomplish during the events of AoA? What are their overall goals?
Intervene in the development of the Terrans and Vasudans to keep them on course for the role the Vishnans wanted them to play: as worthy inheritors of the Brahmans. The Vishnan intervention altered the invasion of the UEF and brought the Sanctuary survivors over to the 'prime' universe.
All Vishnan intervention in our worlds is deeply limited by their faculties. When interfering at a particular set of coordinates in space-time, they are required to use only information that's more or less local to those coordinates — they can't bring information in from far in the future when they instance at a specific time.
Did the great darkness destroy the Brahmans?
Tough question. One answer would be that the Brahmans destroyed themselves to avoid the Great Darkness. Another would be that the Brahmans became the Great Darkness.
Any "chance" of a final battle between the two "big ships" (AoA)? 
Yes.
Will we get the chance to take out any more of the major antagonist TEV assets? Atreus, Imperieuse, Serkr team being the obvious examples but a certain Vasudan admiral also comes to mind.
Definite yes on the Imperieuse. Probable no on the others, just to control the branches that have to be accounted for in BP3 — although by the end of that story, who knows!
Is a TEV victory in the war a definite FED defeat or will the finale be a bit more ambiguous than that?
The Tev victory conditions do not entirely exclude some of the UEF victory conditions. And different factions within the UEF will have different victory conditions by that point.
What is it about Nagari tempering that enables the formation of the UEF and council of elders, which is hinted to be an unnatural development?
The fact that the Elders have been in communion with the Vishnans means that they are able to make decisions which guide their society towards enlightenment and eventual union with the Vishnan Great Psyche. It is, so to speak, 'against the rules' for the Vishnans to interfere in the development of species this way, but the Terminal Protocol is failing, the universes are running out of viable places for life to grow, and something has to be done.
Who exactly are the Vishnans, and why do they exist?
The Vishnans are the post-physical intelligences who have passed through the 'filter' of the Terminal Protocol, emerged as cooperators, and been enlightened into the Great Psyche — a redoubt in subspace that is causally isolated from the quarantined remnants of the noosphere. Some of them may be Brahman survivors.
How did Laporte end up playing such a seeming pivotal role in the conflict, and what happens if she were to die?
Laporte has been engineered to provide the Shivans with a capability they can't provide themselves. If she were to die the Shivans would lose that capability and would need to fall back on more basic methods.
I've always been curious about how much control the Elders are meant to have over the UEF, or how much of their power you'd imagine they tend to use?
The system on paper sounds almost dictatorial in that the Elders are unelected and seem able to overrule decisions at a whim, yet (from what is seen) the UEF people's feelings for them range from adoration to at least great respect, and for all their power they seem very hands-off and distant about the actual running of the Federation (which runs with elections etc IIRC). That diplomacy mission is the one occasion you see an Elder travel; would an Elder taking to the field like that be a rare event, or would tours of the Federation by Elders be quite common (assuming they can overcome their lack of mobility, if I remember a bit of text correctly)? What sort of relations do Federation civil servants and politicians have with the Elders?
I don't want to step on this too hard, in terms of closing out creative range, but my feeling is that the Elders are basically grandparents to the UEF, if the UEF were a healthy adult: you don't
have to listen to them, but you're on good terms and you respect them so much that you'll give their opinion a lot of weight. Plus they have money, and probably political strings to pull, and a legally ordained rule as a long-term steering committee — and they're good people who have earned the public's love. And as things have gotten more intense, strategically, the Elders have been forced to be more and more heavy-handed.
What sort of things would you imagine Byrne being taught on his 'Philosophy of War' PhD (other than the obvious 
)? The objectives? Role of motivation and reasoning?
That war must always be considered as a means to a humane end, or else war is not worth conducting.
Given their seemingly peaceful ethos, why does the UEF have so many frigates and cruisers for protecting a single system? Is the GEF so powerful that a large fleet was warranted for a single star system? Was it in case the Shivans attacked? And I've always noticed the UEF seem to never deploy sentry guns; was there any reason for that or just a coincidence and they are in UEF use in canon?
The Shivans might come back! Sentry guns aren't super useful for defending against 'soft' infrastructural attacks like sabotage, bombings, misdirection, and even theft by unarmed noncombatants. The UEF fleet is probably small in proportion to their economy: consider that their economy matches the entire GTVA, then compare fleet sizes...
