I plan to give a talk on Steenrod algebra in a student seminar. but there are some questions that I didn't find an answer to, and it seems to me that I'm missing something:

~~if the algebra of cohomology operations $\mathcal{A}\left(E\right)$ is isomorphic to the cohomology algebra $E^*\left(E\right)$, why it isn't (graded) commutative?~~the right and left units (source & target homomorphism) on the dual $E_*\left(E\right)$ are induced by the morphisms of spectra $\mathbb{S}\wedge E\to E\wedge E$ and $E\wedge\mathbb{S}\to E\wedge E$. why they are usually distict? and why they coincide in the classical cases ($H\mathbb{F}_p$)?

and another related question:

how by "correcting" the non-commutativity of cup product (in spaces or chain complexes) the stability is automaticly "corrected"?