Can you put a description in the OP that tells us what our character looks like?
Done.
What information is available to arrancars regarding the Soul Society, the soul reapers (including current members), and the World of the Living?
Soul Society is the hated eternal enemy of Hollows. It is known to be made of a number of Divisions, with a scale of power based on ranks and Captains that hold terrible power supposedly matched only by the Vasto Lordes. A few names spread across Hueco Mundo, although information tends to be distorted or unreliable; Yamamoto is a terrifying figure of legend, for instance, but you'll find Hollows who still think Urahara is a member of Soul Society. In theory the three shinigami defectors could provide extended information about Soul Society but so far they've mostly shared it with the Espada and nobody has really cared to give Numeros comprehensive briefs.
The word of the living is a dangerous and pointless place. Menos Grande only have dim recollections of their time as humans, no longer need to feed on them, and instead need to feed on Hollows, so they have little interest in going there; and it is infested with shinigami besides.
What point are we at in the timeline?
Aizen has been back for a couple weeks.
Are we capable of going to the Soul Society?
An Espada's Garganta could take you there, but you have no autonomous means of reaching it.
How much autonomy do we have? Are we allowed to go to the Living World and find soul reapers to kill without permission from the higher-ups? Can we rampage through the Soul Society if we wanted to?
You're actually officially forbidden to interact with shinigami to avoid playing Aizen's hand to early. That said, there's an informal understanding that this a guideline more than a hard rule. If you go to the living world on superiors' orders, a shinigami finds you, and you kill them and tie off any loose ends that could reveal you, they will let it pass. If it transpires that you're actively
looking for soul reapers to kill, though, you'll be disciplined. You're also not supposed to go to the living world without orders. Recall that the exact plan you propose got Grimmjow kicked out of the Espada in canon (this hasn't happened yet in this Quest, if it happens at all).
Even assuming you could open a passage to Soul Society, get there and "rampage" and not die within half an hour at best, you would most certainly get executed for that terrible mistake once you come back to Las Noches.
Considering you need to have fought and killed a lot of Hollows to survive long enough to become an Adjuchas, as we have, how come we're only Adept at Combat Arts? Is it because we're unused to our new body?
You've always been physically weak for an Adjuchas, relying instead on your special abilities and powerful Cero. Note that even so, as a Combat Arts Adept, you're still stronger than ordinary Hollows even when only using your unreleased sword.
Do bodies disappear quickly only if the killing blow is struck by a zanpakuto or if they're injured at all by a zanpakuto?
Answer unclear at this time. Bleach canon is inconsistent here, with various characters regularly staying around as dead bodies despite being killed by a shinigami's sword.
Can we fire Cero as fast as Grimmjow at Skilled Cero? What colour is our Cero? Are our balas the same colour as our Cero or are they red?
Quoting the winning write-up:
"
Your Cero and Bala are pale, white bursts of wispy power released from the depths of your soul; almost wraith-like in their silver-pale and dreamy glow, they dazzle and confuse opposition; a smokescreen for their destructive power. Of special note is your ability to perform the Cero Fantasma; an eruption of wraith-like energy projections around you that quickly anihilates any nearby structures and enemies while playing on the heartstrings of opponents with the keening screams of mourning that they exude. "
You can't fire quite as quickly as Grimmjow. Your wind-up speed is fast enough that you could use your Cero to cap off a series of attacks that left an opponent stunned, but not enough that you could pop it in the middle of a series of attack.
Which Numero was the last to be arrancarised without the Hogyoku?
Answer unclear at this time. You were one of the very last, so it'll be someone in the #49-54 range. Unlikely to be a canon character unless those numbers happen to be filled.