Charlotte names the spider Sting after seeing a copy of J.R.R. Tolkein's The Hobbit. Sting is not the name of a spider in the Hobbit, rather the sword Bilbo uses to fight the spiders.
Charlotte's name is more than likely a reference to Charlotte's Web, another story about a clever spider.
When Ethan asks Frank why he carries a nail gun while he is an exterminator, Frank answers "protection". The nail gun is likely a reference to Arachnophobia (1990), another movie about carnivorous spiders where the main character finishes one of the creatures off with a nail gun.
In the basement, when Frank sees Sting's blood and says "If it bleeds, we can kill it", this is clearly a nod to Predator, when the classic line is spoken by Arnold Schwarzenegger.
In the basement, when Sting creeps up to Charlotte and she turns her head away in fear towards camera, this is extremely similar to a scene in Aliens 3 between the xenomorph and Ripley.