Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$
Your Conjecture 1 is indeed true. It can be proved using diamond lemma techniques and the proof is sketched in Section 2.1 of the paper
Bergman, George M., The diamond lemma for ring theory, Adv. Math. 29, 178-218 (1978). ZBL0326.16019.
Essentially, using the diamond lemma, it turns out to suffice to check that the two ways to reduce $ccab$ and $cabb$ using (either by first reducing $cc$ or $bb$, or by first using your identity for $cab$) are equal, and this is a straightforward computation.
(This answer was adapted from darij grinberg's comments on the question.)