Can you take companions into dead money?
Accordingly, what should I bring to dead money?
You can't bring anything into Dead Money. Leave everything at home once you unlock the fast travel point for the bunker, because you'll need all the carry weight you can spare when you finish. Don't go into the bunker until you're absolutely ready, because you can't leave until you're finished.
Then, can you bring companions to Sierra Madre?
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.
In fact, each of the DLC except Old World Blues have sections in the main quests that require you to have a companion in one form or another, as well as a lot of sections where you can bring a companion, but it isn't mandatory.