Did h2 and h3 change the phonetic reality of an adjacent *e in PIE under any circumstances? Can we treat *a and ā as allophones of *e in PIE?
By definition yes. They're called the a-colouring and o-colouring laryngeals entirely because of the effect they had on adjacent *e.
Denying phonemic status to *a/ā is not universal, but it is done by the Leiden school, who analyse every *a as *h₂e and every *ā as *eh₂. This leads to a reconstructed language with very few vowels, but there are decent reasons to do it: *a would be unusual in that doesn't participate in ablaut alternations (like *e, *o, *ē, *ō do), doesn't alternate with a glide (like *i/j and *u/w), and doesn't show up in suffixes or endings, and it just generally isn't needed.