I am new to genetics and I know humans share many genes with mice for instance but that there are slight differences in conserved nucleotide sequences. Is there a community consensus around at what point a gene in one species is a different gene? How are regulatory elements like enhancers factored into this definition?
From an evolutionary angle, you are describing an ortholog: "Orthologs are genes in different species that evolved from a common ancestral gene by speciation, and, in general, orthologs retain the same function during the course of evolution." quoting here. A lot of effort has been made to construct ortholog databases (e.g. see quest for orthologs). They are distinguished from paralogs (approximating duplicated genes) and lateral gene transfer events, there is however, no magic cutoffs for defining orthology.
Your 2nd point about regulatory elements, well they could fit in the same framework, any sequence can be clustered. What is a gene anyway? Is a noncoding RNA a gene? How do we go from a collection of transcripts to this unit we term gene? There is some ambiguity...