I was wondering why some contractions in sentences don't make sense in their unabbreviated form, such as "why don't we do something about it" versus "why do not we do something about it"
You can get close by changing the word order slightly: "Why do we not do ...". Once phrases, contractions or acronyms become established, though, they tend to continue their linguistic development somewhat independently of their roots.
In the case of "Why don't we X", although it can be understood as a request for clarification (i.e. Why do we not do X?), it can also be construed as a suggestion to do X, without any hint of a request for clarification.
Why interrogative adverb 1.1 [with negative] Used to make or agree to a suggestion: ‘why don't I give you a lift?’ - ODO
Interrogative sentences, both closed ended (polar) and open ended can be in the negative. In Wh-negative questions ('who' excluding) and yes-no questions inversion of auxiliary verbs / modals before the actual subject is mandatory (after wh- words if any). Inversion of NOT is optional. The short form of NOT is n't. As a general rule we may say that if this short form is used, we invert the shortened n't with the auxiliary or modal, otherwise NOT is retained at its usual place.
We were once tought not to invert 'not' in its full glory if the subject is a pronoun; no matter if a noun. But I haven't come across any authoritative source as yet.
Actually, one can find the construction "do not you agree" in (for famous example) Jane Austen, but I guess we'd call it 'archaic' when we encounter it in speech today.