I agree with janise and pjs. Go straight to Nice on arrival and keep all of your Paris time together. That will save you 1 day of checking out, travel, and checking in twice.

Yes with janis - fly to Nice first - if going from Paris center trains are not all that slower than planes and more likely not to be delayed. Or of course spend all Paris time first and fly home from Nice via Paris or London or whatever is best.

Agreeing on a several points:

Fly into Nice and Out of Paris. If coming from USA, there are some very efficient connections to Nice via Madrid or other major airports that get you to Nice by mid-day.
There is plenty to do in Nice that will fill 2.5 days...no day-trips needed unless you really want to.
Paris has so many authentic things to do, I'd avoid Moulin Rouge.

ssander

Thank you everyone. I plan to move our time in Nice to the beginning of the week.
Stu - I appreciate the Paris tour recommendation as well as the itinerary for Nice.
5 alive - Thank you for the hotel and museum recommendation, We will definitely check it out.
MJS - We would like to stay at $300 or less/night for hotel budget. We are open to suggestions and appreciate any input.