Given their origin in mythology and legend as guardians of sacred places, shouldn't gargoyles be lawful good? 

They're inspired by real world history, but in D&D lore they're instilled with hatred even before their form finalizes on the elemental plane of Earth.

Quote from NordicNinja >>

They're inspired by real world history, but in D&D lore they're instilled with hatred even before their form finalizes on the elemental plane of Earth.

 
I should note that this isn't true on Eberron, where their origin is disputed (there's no Plane of Earth for starters) and they, as mortal Elemental creatures, have no set alignment (though are likely to be aligned with "monstrous" factions). Additionally, on Eberron I would suggest a History check instead of RickAJr's suggested Religion check, because they're more part of the political landscape of the world than the religious one (with the weird exception of Wingwyrds). Entirely unasked for information, I know, but I enjoy this stuff. :P

Without Alignment, here are some ideas of their motives:
They're pretty likely to be Agents of Droaam in the employ of the Daughters of Sora Kell, so mess with them at your own risk. If they're in Sharn, they're likely doing so as part of the criminal organisation Daask, and using their False Appearance trait and power of flight to full effect in the vertical city. Bear in mind that they can fly and you are probably fighting them from a thousand-foot-high bridge, so make sure to stay away from the edge. They could also be bodyguards for an ambassador from Droamm, likely a medusa, and may not even approve of Daask's methods, but I say that based on the fact that they generally live among the medusae and show loyalty to Droamm, not because I'm aware of any distinct examples.

Of course, if you're in Sharn, they might just be civilians.  You could also see a Gargoyle postie or detective (as they are often employed by House Tharashk), so a suspicious-looking statue might be doing the work of the law...

... Or, perhaps, you're very lucky and getting to meet Carralag, the sporting celebrity who competes for Malleon's Gate (and Fallen, but probably nobody cares) in the Race of Eight Winds as the only riderless competitor. He may or may not be funded by Daask, but he's mostly just a cunning racer so there's no need to cause trouble; just be sure to get his autograph (and maybe sell it to some of his goblin fanboys). If there is trouble, he (in my opinion) likely doesn't know it; it'll be fanatics or Daask (or both; there's overlap) trying to protect their investment. Better not to implicate the talent.

Quote from GWeber1 >>

Given their origin in mythology and legend as guardians of sacred places, shouldn't gargoyles be lawful good? 

I’m glad they reduced the number of attacks they make.  It used to be something like claw claw bite horns tail for a total of five attacks. Regardless of combat balance this totally slowed gameplay to a crawl, especially if there were multiple gargoyles. 

Quote from GWeber1 >>

Given their origin in mythology and legend as guardians of sacred places, shouldn't gargoyles be lawful good? 

 Fun fact, one of the first instances of gargoyles being used in literature was in the book "Maker of Gargoyles" where the a stonemason was abused and disrespected by the people, and clergy, of a town where he had been commissioned to make gargoyles for their church. Unbeknownst to him, he imbued the first gargoyle, a winged cat-faced creature with mighty claws, with his rage, and the second, a satyr with bat-like wings, with his lust and desire for power over the citizens. They would come to life at night and enact his revenge, eventually ending in his own murder when he discovered their work and confronted them. Fascinating read.

:)

Does anyone happen to know if gargoyles have any natural enemies or if any part of them is worth trying to capture or kill them for alchemist material?

@Amonon Demoneye

Well for natural enemies it does say that Aarokra are enemies with them with the quote

"these birdlike humanoids guard the windy borders of their home against invaders from the Elemental Plane of Earth, such as gargoyles, their sworn enemies."

Thanks!

Aarakokra are Gargoyles natural enemy

Food for thought: an earth elemental has damage vulnerabilities to thunder, gargoyles are elementals (made of stone), why aren't gargoyles vulnerable to thunder damage as well?

I'd say because Gargoyles resemble statues, Regular Earth Elementals are more like a Walking heap of Boulders. In other Words an Earth Elemental gets shaken into bits while Gargoyles are a Solid single piece less effected by Tremmors. Sort of ironic if you think about the Earth Elementals Tremmorsense. 

Does anyone have any interesting suggestions for playing these guys in combat? Considering their apparent love of cruelty I was considering them choosing to grapple characters and drag them upwards with flight before dropping them (directly onto something hard, like a tombstone). Any other suggestions that could get their personality across?

Just curious but Stone has an AC of 17 and other stone creatures like Stone Cursed, Earth Elemental, and Stone Golem also have AC 17, any reason the Gargoyle is easier to hit?

If I raised the AC to 17 would that raise the CR to 3?

I just play them as any lawful alignment.

Is'nt adamantine already magical?

Adamantine Armor is a magic item that happens to be made of adamantine, but the metal itself isn't inherently magical.

The core D&D books (Players Handbook, Dungeon Master's Guide, and the Monster Manual) don't actually make any references to adamantine weaponry besides certain monsters like the gargoyle being vulnerable to it, but Xanathar's Guide to Everything has price guides on how much non-magical admantine weapons cost, and what effects they have.

Read "The Stones of Muncaster Cathedral" (Robert Westal, 1991) for lawful evil backstory ideas.

That book has some epic plot hooks for an early level campaign.