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To be honest - it depends on the Airline - that's why I suggest you call and check with them or better yet check their website for the information . . .
Now you know I do not have any children - so this is all from what I have seen other do when flying and also only my opinion based what I know . . .
So - Personally, given your reports about "Levi" not sleeeping anywhere else - but with you both, then I wouldn't be bring any crib (unless it is one of those softsided folding ones), especially since you could make one out of a few pillows and a blanket/quilt when you get to where you are going. As for the stroller (hopefully, again foldable) and the car seat - you maybe able to bring both on the plane (just in case there is an empty seat next to you too) - however, if they don't allow them once you get to the plane they usually will check it right there for free and then bring them right back to you when you are getting off the plane . . .
Also - Most Airlines (not all I know Jet Blue lets you check 2 bags for free) charge for each bag checked $ 25 and up for each additional bag checked - they must also be NO MORE THEN 50/70 lbs each - depending on Airline . . . Each passanger is allowed 2 carry-ons (size limits for these agian depending on Airline), usually a small suitcase (for the overhead bin) and weekender size (that will fit under the seat) . . . Women's handbags do not count, but totes/breif cases/etc. do . . . As an example - I bring on the plane a med size suitcase full of clothes and extras & a "Backpack" filled with stuff I do not want to lose that includes Meds/Personal Papers/Electronic/Emergency Food/etc. (it also has wheels) . . . You could possibly get away with a lot of things though without checking stuff if you think about it, epseicllay since there are going to be 3 Adults and 1 baby . . . Again, check with the Airline you are flying either on their web site (the best) or call them to find out what is allowed/not allowed on the plane and any possible additional cost chacking stuff . . .
Oh and DO NOT FORGET to bring little paper notes apologizing to your "Seat Mates" with a candy/snack or something special like ear plugs to say this is your son's first flight and your sorry if "Levi" get's out of hand and crys all the way to Florida . . .
(Someone actually did this and asked the flight attdants to hand them out - the results was everyone on the flight thought it was a hoot and laugh - of course forgiving the parents . . . Just Sayin' . . .
)
Sorry
- but it was funny . . .
VERY GOOD IDEA 
