The Tuileries is always one of our favorite parks, but during the holidays, it's even better. It's right next to the Louvre, and it has the L'Orangerie museum (small, very doable--the one with all the Monets), and you can take have a hot chocolate treat at Angelina across the street. One of our other favorite lunch spots, for things like Croque Monsieur, perfect hot dogs and fries...is La Coupe d'Or. Their sandwiches are made with the famed Poilane bread, and they also have the renowned ice cream Berthillon. (330 Rue St Honore). If you're looking for a breakfast option, we love breakfast at Cafe Marly, which overlooks the pyramid of the Louvre. It's not cheap, but it has the best view. (93 Rue de Rivoli). While you're in this neighborhood, Colette is worth a quick look...it also has a very cool cafe on the lower level, which has loud music, very cool hi-chairs, and shows movies on a screen. Back to the Tuileries, the ferris wheel was going when we were there last year, and that is a worthy bribe to keep in your back pocket. The playground is pretty epic, and depending on the weather, there could be the sailboats in the fountains to push around. And then there's the carousel, and the trampoline, which is awesome for getting them warmed up or burning on steam. At the base of the Tuileries is the Champs Elysee, and at this time of year, there are tons of stands selling delicious food...sausages, foie gras, sweets...and then there's a cool trampoline with a harness that kids can do, and a super long slide.
Eiffel Tower Visit: We found it so annoying to try to get to the top--the lines were so long...so unless that's really important to you, don't even offer it up as an option. There's a great playground at the park, and there are several carousel options across the street from it. And maybe you get them a tacky battery operated Eiffel Tower toy which would be so much cheaper than paying to go to the top. In that neighborhood we liked the Quai Branly Museum which has a fantastic outdoor garden, which you don't have to pay anything to enjoy, and we liked the Palais de Tokyo, which I remember mostly for having a very cute, modern gift shop and restaurant. 
Merci: This is one of the best shops I've ever been to, and it brings you to the Marais area. They do have a cafe there, but it isn't really where I'd spend my lunch, unless there was a kid crisis and I had to make do. Plan ahead to spend at least an hour walking around Merci, and then just down the street is the out-of-this-world ridiculous Bonton superstore, which is for kids. Send one parent downstairs where there's more of a hangout area for the kids...and wander around upstairs, not necessarily because you're going to buy anything, but more because it's so inspiring. Before you leave, check out their photo booth--last year they had these Christmas themed cut-outs to frame your face in, and we're using the pictures from then for this year's Christmas cards. In the area, the two special places we ate, that weren't the kind of places that you'd have to wait in line for, are very typical cafes: Cafe Charlot (I particularly remember the tarte tatin), and Cafe Janou. (If you go to Janou, order the chocolate mousse for dessert--it comes in a huge bowl and you help yourself to as much as you want.) For warming up and getting out of the cold, and away from retail, I really liked the hunting museum, the Musee de la Chasse. The Picasso Museum is also in that neighborhood. Stay away from this area on Saturdays if possible, when it's annoying just like any shopping district is. 
Sunday in Paris: Not everything is open on Sundays, so we opted to go to the flea market in Vanves, which is in the outskirts of town, and has some great finds. There are no easy bathrooms, and there isn't much else out there, so you might send the antique obsessed off on their own and not subject the kids to it. We somehow skipped Clignancourt, which I mistakenly thought was open on Monday (it's not), and took a lot of heat for. So if antiquing is important, know that Clignancourt is best on Saturdays, and fine on Sundays. Because there are good food options there, and indoor spaces, I would say you could take kids to Clignancourt. Moving on from antiques...one of my favorite finds for a Sunday in Paris is to go to the bird market, on Ile de la Cite. The birds are so magical, and when have you ever been to a bird market in your life? We learned about it from reading Sasek's This is Paris, which could be your guide on your whole trip. We also checked out Notre Dame that day, and walked around L'Ile Saint Louis. If you are in that area, a stop at Berthillon, the ice cream shop, is a must. And across the street from it is a little food shop that has the best raspberry jam I've ever had in my life. I bought 5 jars there last year. (Yes, that was a hint.) 
Jardin d'Acclimatation: A super cute old school amusement park on the outskirts of town. This is at least a 3 hour commitment, if you factor in time to get there. There are fun rides, cotton candy, what you'd expect...but it's Paris, so it feels different and better than what you grew up with. (I have no ideas about anything that's nearby.)
The Left Bank: For me the first stop on this side of the river is the a store. Yes, I'm a sucker for little girls' clothing, so one of my favorite spots is the Bonpoint outlet. No, it's never cheap, but it is hugely discounted. I go for their socks, which are so well made that there's never a hole even in the socks that last her 3 years because I buy them big. Beyond that, I love the Luxembourg Gardens park...another great playground, but if I'm remembering correctly, you have to pay to get into it. There's also pony riding there. And often a puppet show. If you're super jet lagged and can manage to wait til after 2pm to eat lunch, you should go to Le Comptoir. It's a favorite amongst all Parisians, and anyone who goes there. They don't take reservations, so it really is a spot to only consider for the hours of 3-5pm. Also in that neighborhood I like to stop at an Andre shoe store (our equivalent of a Steve Madden I guess), because their shoes are cheap, and excellent knock-offs. I also love the store Talc, which is on one of my favorite streets, Rue de Quatre Vents...it has such nice kids clothes, and they're not as astronomical as Bonpoint or Bonton. But if you want to stay away from that, at least check out the sweetest little wine/cheese bar, La Cremerie. 
Le Bon Marche: It has such a great selection of all the Parisian designers, plus a great food court, and, from what my friend told me, great kids section. It also has a park right out front.
Places I would recommend, but aren't fitting into my neighborhood pairings:
Monoprix: the Kmart/Target of Paris. They have great kids pjs, and lots of cute clothes for kids that are made with organic cotton. Well priced.
Lunch or dinner at Le Grand Colbert. 4 Rue vivienne in the 2nd. It's super noisy and boisterous so even with a meltdown, if should be fine. It's right within an arcade that has a super cute kids toy store.
Dinner at Le Bistrot Aux Lyonnais at 32 Rue St Marc.
Dinner at Robert et Louise in the Marais, where they grill steak on an open hearth.
Buying loose tea (Wedding Imperial is delicious) and a canister to keep it in once it's home, from Mariage Freres.
Shopping for yourself at Cos, which is the high end H+M, in the Marais, and has the best best clothes, for very good prices.
One of the best online resources for Paris is Babyccino Kids. This is highly edited information, and it certainly isn't covering where to buy tickets and the best way to do whatever...but if you're staying in an area and want to know the best cup of coffee and kids shop nearby, check it out.
