The bulk of the spider effects were done practically.
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 Aracnofobia (1990), another movie about carnivorous spiders where the main character finishes one of the creatures off with a nail gun.
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.
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.
Sting the spider is very much like the Australian red back spider which is one of the most venomous spiders.