Very short answer The phrase quelque chose is invariable.
(Short answer: see the end of this post, after the long answer.)
Long answer The catch lies in the fact that quelque and chose can both be used by themselves in other contexts, where they might need to agree in number.
Let's tackle the simple one first: chose by itself means thing, and is just as vague and "catch-all" as thing is in English. Une chose = a thing, des choses = (several, some) things. See here if you want some more details.
The other simple word to deal with is the plural form quelques, which means a few:
J'achète quelques pommes = I am buying a few apples
Avec quelques amis = With a few friends / With some friends
The singular quelque is much more complicated and I'm no linguist, so I'll try my best. When related to its plural counterpart, quelque is usually formal and basically means some / a little:
Avec quelque difficulté = With some difficulty (vs. Avec quelques difficultés = With a few difficulties)
Quelque also has a couple other formal meanings that I'm going to ignore on purpose here for the sake of clarity (However + adj, About / Around). What we're interested in here is the few very common fixed phrases that use quelque. These aren't formal, and are invariable:
Quelque part = Somewhere
Quelque temps = Some time
En quelque sorte = So to speak / In a way / Sort of
And of course:
Quelque chose = Something
Ok, with all of this in mind, I am going to be able to make my point below.
Short answer Quelques choses would be grammatical, and would mean a few things. However, the fixed phrase quelque chose is so common that any native ear, upon hearing quelques choses, would process it as quelque chose (they are pronounced exactly the same). So if you actually want to say quelques choses, you would need to be more precise to be understood. This can be done by changing choses to a less general word, or by adding an adjective in between:
Quelques petites choses = A few small things
Quelques objets = A few items
I hope this clarifies a few things (huhu). [Also quelque is no longer a word to me :D ]
According to larousse it seems that in a question it could mean "anything" depending on the context, http://www.larousse.com/en/dictionaries/french-english/quelque/664448#64875.
Because in English "do you want to eat something" and "will you eat something" mean very different things. The first is asking basically this: "Do you, at present, have a desire to eat something?". The second is asking basically this: "Will you, in the future, eat something?"
I'm not an expert in français grammar, but I'm thinking you'll be immediately understood if you write "une femme francaise", or "a la carte" and if you tell a server "I don't want no peanuts", you are very unlikely to be served peanuts; but none of those are grammatically correct, either.