Charlotte's name is more than likely a reference to Charlotte's Web, another story about a clever spider.
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.
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 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.
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.