(Long post incoming)
Spideraxe's suggestion is pretty accurate.
I'd probably start in the Piltover/Zaun area. They are very poorly defended and don't even (to my knowledge) have a military. Plus, Zaun is in a constant state of infighting so don't expect an organized defense force there.
From there, I'd allocate all my resources into making it a defensible location (possibly inventing some kind of hextech barrier around the city?) and developing the latest technology for warfare purposes.
Then I'd go east into Bilgewater.
With how lawless it is anyway (and also, surprisingly, possessing no military), it should be easy enough. Plus, the technology in Bilgewater, although advanced compared to other nations, still pales in comparison to the hextech in Piltover.
Since I'm starting out in the smaller places, it won't raise as much suspicion and the other nations won't begin to make defensive measures until later.
With both Piltover and Bilgewater under my control (which are both coastal maritime nations), I now both have the most sea access out of everyone else, AND I also control a majority of the trade, meaning my coffers are always full to fund my weapons research as well as my war campaigns.
Then I'd probably go further east into Ionia.
They aren't a militaristic people so they don't have a large army, and they are certainly lacking when it comes to technology (that the Noxians didn't possess during their invasion) so it should be fine.
The only one that might be a problem is Syndra, although a couple of rockets should be able to bring down her floating fortress (and her with it).
That was the easy stuff though.
Then I'd sail from Ionia all the way around Noxus into the Freljord.
I have the utmost faith that the warring barbarian tribes stand absolutely no chance against advanced hextech technology. The only ones that might be a challenge are the demigods there and Lissandra but the rest should be easy pickings.
Unite them all under one single banner and look towards weaponizing True Ice (perhaps combining it with hextech?).
After taking over Freljord, Demacia and Noxus might start to take defensive measures, but due to their hatred of each other, I do not expect them to unite.
Then I'd go into Demacia.
Although incredibly defensible, since I'm predominately using technology, their anti-magic capabilities are ineffective.
Now, after surrounding it on all sides, I go into taking over Noxus.
The greatest strength of Noxus is it's numbers, but now, with both Piltover, Zaun, Bilgewater, Ionia, Freljord, and Demacia under my control, their numbers mean nothing.
Most of Noxus' outer territories would probably also be inclined to rebel against them.
Now, with the entirety of the north under my control, I push south.
Shurima and Targon might look into creating an alliance since they are close to each other, have close ties from the past with Targonians helping the Shurimans, and both of them have incredibly powerful magical beings.
So for now, I'd invade the Shadow Isles.
Since it is said that silver can harm the creatures of the Shadow Isles (and Demacia has silver in abundance), I'd make sure that most of my weapons are made out of it.
After taking over the Shadow Isles, the greatest challenge of all comes. Trying to take over the Shuriman Continent.
I start off by taking the northeast. Kumungu is uninhabited, and the north was already under the control of Noxus prior.
Icathia is of no interest currently, so it is ignored.
The first major targets are Nashramae and Kalamanda. Taking both down will allow complete control of the north and a safe route from Valoran to Shurima.
With the north-east under my control, I try to take over the far south. Marrowmark, Zirima, etc. etc.
After that, by surrounding it on all sides, I try to take over The Sun Disc.
This will be tricky as Azir is a powerful being, but hopefully Demacia's anti-magic capabilities could come in handy.
After taking over The Sun Disc, I mop up the rest of Shurima and then go into Targon.
Targon is incredibly defensive given it's remote location on top of a mountain, and their Aspects are powerful. If they sent Aurelion Sol, I'm pretty much done for. It might be better to try the diplomatic route and maybe even be a bitch and actually submit to them instead of them having to submit to me. After all, the only thing Targon really wants is to defeat the Void.
After either getting Targon to submit (or submitting to Targon), I've conquered Runeterra (most of it), and the only thing left to do is venture into Icathia and defeat the Void, although that is an entirely different subject.
So yeah, that's how I would conquer Runeterra.
Really the only flaw I see in this is the early invasion of Freljord, but that's mostly because the power level of the demigods there aren't specified. I don't know if advanced hextech technology is enough to take down a god-like being.
As for your questions:
(Since question 1 and question 3 are similar, I'll just merge them together)
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Most vital position is Piltover. It's the center of everything, has sea access, controls the trade, and is technologically advanced.
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Hardest places to try and take over would be Demacia, Noxus, Shurima, and Targon. Noxus made Ionia look hard to take over, but I don't actually think it is that hard with the proper technological superiority. The only alliances I foresee would be one between Shurima and Targon. I don't think anyone else would ally.