1.a - What was it like?
Based on the information we have, it is likely that the Isles were similar to Ionia - a place of spirituality and magic, housing varied races with various skills.
Info from one of Bard's reveal suggest that the Isles fell to ruin very long ago. So it is also possible that Ionia owes part of its heritage to the people that fled from the death and darkness.
1.b - Did they have an arena? Did the kingdom have enemies? Why was an Army needed?
Noxus definitely has an arena. I don't think it's far-fetched to assume Demacia would have one as well since it is also a pretty militarized empire; it might be a little more orderly than Noxus, like duels between noble houses or magical contests instead of the gauntlet-type stuff you'd see from Noxus. So I don't see why the Isles wouldn't have had one in a different form.
Every kingdom has enemies. Kingdoms need armies for when diplomacy fails... or just to further the goals of the empire; depends really.
1.c - Who was the precious one to the king who died?
Caveat - The following is based on old lore from the Shadow Isles Twisted Treeline update and may no longer be true
The King's daughter, and his only heir if memory serves, was the one killed. His grief over this this incident is what eventually resulted in the Isle mages being commanded to tamper with life and death; presumably their orders were to do whatever, anything and everything it took to bring her back.
Notably, Mordekaiser was confirmed to be the very first of the 'undead' that now roam the Isle shores. Lore at the time stated that Mordekaiser had somehow betrayed the King; this is potentially referenced by the West Altar spirit's warning to Hecarim about trusting Mord. However, the nature of Mord's betrayal of the King was not explored.
Perhaps the betrayal was breaking the barrier between life and death, inadvertently unleashing the Black Mist and dooming the Isles.
Perhaps the betrayal was twisting the intent of the King; the King may have sent Mordekaiser and a group of mages to the heart of the Isle's magic in some forest in order to plead with the local spirits for some way to save or return the Princess, but something happened due to Mord that resulted in more death and darkness.
We do not know. Just like we do not know who killed the Princess; we know it wasn't Mord.
Caveat ends here
Couple problems with this old story:
1. If Mord did betray the King, he should be on Kalista's hit list; as far as we know, he isn't, but Hecarim is.
2. We know that Mord can 'curse' others simply by touch, turning them into undead of sorts upon death. But was Mord responsible for spreading the undeath or the Mist? Or was it a side-effect of the life-death cycle being broken?
3. Mord's VO references sickness a lot, which might be closely tied to the Isle origins. If that is the case, it is likely that the sickness he references is metaphorical instead of literal (i.e. life being a sickness with death/undeath as a cure). But then again his VO is ancient so...
2 - After the necromantic explosion, what has become of the Isles? What are the inhabitants like?
The Isles probably look like a darker, slightly more run down version of its former self. If we look at Twisted Treeline + background from Kalista's login, and apply that to an entire island, the whole place doesn't look bad; well... in a manner of speaking. Structures still stand, the land is still there. There are just more cracks, spectral blue/green lights around and the inhabitants aren't so friendly. Pretty much everything you'd expect from a fully-functioning society of undead.
The last Harrowing video with the puppeteer and general descriptions hint that the Isles are full of Shades. Dark, mysterious twisted versions of the creatures they once were. Lucian's login page is probably a good reference for what the humans have become. I guess Hecarim can be a reference for centaurs but we don't know for sure that he was a centaur in life.
2.a - Is the king still in power? Does he still rule?
Maybe. Maybe. That's the next real chapter of the story though isn't it?
Some fun speculation
Maybe he's on the throne. Maybe he passed without becoming an undead.
Or maybe... the Black Mist is the last of his will made manifest; forever spreading outward towards life, consuming all in the search of something or someone that could bring his beloved Princess back. Kalista could be his enforcer/recruiter.
She is called by the betrayed and hunts their betrayers at the cost of their soul. But what does Kalista need a soul for? She isn't a 'warden' like Thresh. But wait, the Sentinels she sends out in-game have faces and are ordered to 'seek'/'be her eyes'. If we apply this to lore, who or what is she searching for?
It sure as hell ain't new souls, we've already established that people offer their souls to her in exchange for vengeance. So maybe, just maybe... she still serves her King after all this time. Eliminating betrayers since the King was once betrayed, all while also recruiting those willing to join her cause.
If Kalista is the King's right hand, and her last order was to exact vengeance on the betrayers that took the Princess and to find a way to bring her back, suddenly her freedom of movement with the Mist and the difference between her modus operandi and the other Isle champions doesn't seem so strange does it?
Fun police, party over
2.b - Is this why hecarim is still patrolling the Isles as he did in life?
Hecarim is patrolling the Isles now, true. However, his current duties were assigned post-mortem.
He did not patrol the shores in life. Neither did he serve the Isles prior to death. The West Altar makes a reference to this, telling Hecarim to remember who he once was.
v.1 - Lucian is still out hunting undead. It is said in his lore that him and his wife were given those guns at a time before undead was reported roaming Runeterra. Does this mean that the isles remain to them selves before they started terrorizing the living?
A lack of an undead presence doesn't mean there would be a lack of 'holy weapons'. Mutants from Zaunite experiments like Twitch, Void creatures, and other dark-creatures (like Tibbers before being enslaved by Annie) could all be justified as enemies that require 'cleansing' as opposed to a more straightforward death.
Also, Noxus has a history of necromancy (Sion once upon a time, and recently again) so holy weapons might've been originally developed to combat that threat.
The Harrowing event map from last time seems to suggest that the Isles might've been in a hibernation state, since all life was consumed on their limited landmass. Think the eggs and drones/warriors from the Alien series. They are active when 'prey' are near but are able to patiently wait once the 'prey' have been exhausted.
Not all undead retain their minds or sense of purpose. It is possible they simply thought all avenues of life had been exploited until a ship got too close. Once a couple of ships got close, the undead got smart and started going out to sea in order to get their 'prey'.
v.2 - Apparently, people can travel to the isles and leave unharmed. Such people as Elise. How was Elise able make a pact with Vilemaw, the undead supposed spider god? So ezreal could go there explore and remamin unharmed?
The Isles, Black Mist and undead champions might be able to sense the would-be-victims' intent.
So people that run scared or fight are consumed then enslaved as drones while people that come willingly are treated differently. That would explain why Elise was able to become the con-artist she is and why Karthus was able to simply give himself over without losing his free will.
Hard to say about Ezreal. We don't know for sure what happens and the above was speculation. Ezreal is known to have latent magical prowess and when combined with the gauntlet on his arm, it might be possible for him to 'resist' (even if temporarily) the corruption of the Isles.