This is a multi-part question, which I know is generally frowned upon, but please hear me out on this.
If you have Veronica as your companion (assuming you can even take companions to The Sierra Madre), does it impact your interactions with Elijah?
Otherwise, do your interactions with Elijah impact your interactions with Veronica, and if so, how? (This would be most relevant to the iteration of New Vegas that I'm playing, as I am currently in the Dead Money DLC and have intentionally not yet met Veronica.)
Best Answer
You don't take companions to The Sierra Madre, and having taken Veronica as a companion before will not impact interactions with Elijah.
After finishing the DLC, you will receive an encrypted holotape that only Veronica can decrypt. Go talk to her, exhaust all conversation trees related to Elijah and say you've found him. Then, you'll have the options of giving her the Holotape or ask her to decrypt and keep it. If you give it to her, she will get the perk Elijah's Last Words (+150% melee attack speed, 25% chance for melee attacks to knock down enemies for Veronica), if you keep it, you will get the perk Elijah's Ramblings(+%150 melee weapon critical damage).