The first paper on abelian categories is "Exact Categories and Duality" D. A. Buchsbaum. It was published in 1955, two years before Grothendieck's famous Tohoku paper. You can find it on jstor. The section 5 is about "fundamental lemmas" such as the Nine Lemma (5.5), the Snake lemma (5.8) and the Five Lemma (5.9). The proofs are direct using the definition of an abelian category (called "exact category" by Buchsbaum, this term was used later by Quillen), in particular they use - of course - no elements. Unfortunately I cannot find the salamander lemma, but lots of basics of homological algebra such as homology, derived functors, satellites. Basically Buchsbaum observed that there is nothing special to module categories treated in Eilenberg-Cartan's landmark book on homological algebra.