By experiment, ChIP-seq, or CUT&RUN. I personally find the entire way of in silico binding site prediction utterly unreliable. Motifs are highly redundant, and motif presence does not imply TF binding.
Finding a binding site for a transcription factor, your Gene1, is almost impossible without experiments unless Gene1 is homologous to a known transcription factor.
One can predict putative DNA binding motifs in Gene2 by collecting promoters of Gene2 homologs and looking for over-represented DNA sequence. But even if you find the motif(s), that won't necessarily link them to your Gene1. The only way to link the found motif without experiments to Gene1 is if Gene1 is homologous to another transcription factor with a known binding site, and that binding site matches the motif you found.
The answer to your question #2 is to scan the promoter against a known database of binding sites, but that will only give you some idea what might be binding there. It won't necessarily provide a link to Gene1.
By experiment, ChIP-seq, or CUT&RUN. I personally find the entire way of in silico binding site prediction utterly unreliable. Motifs are highly redundant, and motif presence does not imply TF binding.