Can I advertise in India for a nanny/cook/personal assistant to work
for me as a live-in in my home in the US and when I travel
internationally?

Sure, you can advertise, but the question is whether you can legally
employ her in this capacity, which may depend on how long the contract
should run.
It may be quite easy to have someone come for a year as an "au pair",
but quite difficult to employ someone requiring a different type of
visa.

Perhaps you can explain a bit more.