1.) What exactly is Malphite? Is he a Golem created by Ancient Magic or is he literally just a montain that became Sentient?
I like the idea of him being a mountain that sprung to life.
"Alien rock person from another dimension" sounds a bit farfetched.
How/Why did he come to life?
How? Probably with the use of the World Runes.
Why? Due to the rampant use of them during The Rune Wars.
2.) Given all the Stories & Lore we're being presented (Void, The Rune Wars, Darkin, etc), what is Malphite's Role in the League Universe?
Being a literal mountain that sprung to life due to the destructive use of The World Runes, Malphite feels the pain that Runeterra would otherwise be feeling. Driven mad by endless, mind wracking agony, he despises mortals for their reckless use of magic, calling them "The Minions of Chaos".
He sees himself as the only one who can restore balance to the world, and wages war in the name of order. He seeks to claim the World Runes, by force if necessary, and use them to restore Runeterra to what it once was, no matter how many mortals stand in his way.
So sort of like Ryze's motives, but Malphite wouldn't trust Ryze. At least not initially. They might get a short story together where they become allies and try to locate the World Runes together. But whereas Ryze would just hide them away, Malphite might try to take them for himself.
His new title would probably be "The Shard of Runeterra" or something akin to that.
Here's some example quotes:
Pick:
"Rock solid!"
Banned:
"You mortals shall not deny me!"
First move:
"Beware, minions of chaos! The Shard of Runeterra has come."
"Pain... Agony... The land weeps and the world heaves with torment. My torment."
"Reckless Mages. I will punish their imprudence. I will end this misuse of magic."
Moving:
"The earth is my shield. My calling."
"I speak for Runeterra."
"Runeterra cries out in pain."
"This world needs me."
"This world must be healed."
"They do not know me... but I know them."
"I am the world incarnate."
Attacking:'
"Time for a mass exctinction event!"
"Feel the fury of the mountain!"
"Enough of your foolishness!"
"I am the voice of the world!"
"Begone vile magics!"
"You will learn to respect the earth."
"There's no shelter from my fury."
Joke:
"I love the way Runeterra rotates. It really makes my day!"
"Yes, I do realise the irony in Runeterra meaning "Magic Earth"."
"Caught between a rock, gods, mortals, The World Runes, the Black Mist, the Void, and a hard place."
Taunt:
"After I have slain all of you, I will create the perfect world!"
"Your tenacity is admirable, but pointless."
"Your efforts are insignificant."
"You are less than dust, fit only to be brushed away."
"You will lose."
Special Interactions:
"You shall not end any worlds as long as I stand." 
"We share equal burdens."

"Mortals are not the only ones at fault here."

"You would abuse magic for something as petty as seduction? Begone, witch."

"You mortals would do well to listen to those wiser than yourselves."

"Do not presume your ancient power exempts you from punishment."

"The world teeters on the brink of destruction, and you would rather play jazz?" 
"Your very existence is justification for all my actions." 
"Voidspawn! Forces of chaos! I will crush you all!"

"Mortals are not worthy of the Aspects' power!"

"Undead. Abominations. You defile this sacred world with your very presence. You are a mistake that must be corrected."

"You know not what hides within your walls... or within yourselves."

"I will not allow this wanton destruction to persist." 
"Every tremor is another thousand headaches." 
"The earth is not your domain, Voidspawn!" 
"Your compassion for these primitive beings disgusts me to my very core."

"Rune Mage! Do you presume that I am at your beck and call? I have a world to heal." 
"The earth is not yours to bend, Stoneweaver." 
This is just my thoughts on a reworked Malphite. A tormented soul that seeks to restore balance and order to a chaotic world, similar to his current lore.