a). Indeed, enolate attacks take place therein and not hydride ones. But it was presented that way due to graphical text savings and simplicity reasons.
b). The retro-Claisen scenario is reasonable. But it demands a strong base; stronger than Et3N or AcO- or even, an enolate.
c). Contrary, decarboxylation via formic carbanion in equilibrium with the enolate that is formed between the three carbonyls, is possible because the equilibrium is pushed to the right via formic acid dehydration that is mediated by acetic anhydride (step 2).
d). Additionally, please note that the high ring strain of β-lactams explains among others, the antimicrobial activity of β-lactam antibiotics in a similar way as in step 3.
1). Recent Advances in the Retro-Claisen Reaction and Its Synthetic Applications, Current Organic Synthesis, 9(4), 488-512, (2012) http://www.eurekaselect.com/100926/article
2). Base-catalyzed retro-Claisen condensation: a convenient esterification of alcohols via C–C bond cleavage of ketones to afford acylating sources, RSC Advances, 4(56), 29502-29508, (2014) https://pubs.rsc.org/en/content/articlelanding/2014/ra/c4ra04618h
3). The catalytic decomposition of formic acid in acetic anhydride, Journal of the American Chemical Society, 45(2), 455-468, (1923) https://pubs.acs.org/doi/abs/10.1021/ja01655a022
4). Carbon-13 Kinetic Isotope Effects in the Decarbonylation of Liquid Formic Acid Assisted with Acetic Anhydride, Isotopes in Environmental and Health Studies, 4(3), 285-289, (1998) https://www.tandfonline.com/doi/abs/10.1080/10256019808234061
5). Advances in the chemistry of β-lactam and its medicinal applications, Tetrahedron, 68(52), 10640-10664, (2012) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3525065/