Our children have their passports. Do we still need to provide birth certificates when applying for an FM3 visa as well?
I don't believe so (although I'm not 100% sure) However, when I studied here, my school required an original birth certificate. I wouldn't be surprised if whatever school you enroll them in requires the same. My university also required various translated and apostilled documents (birth certificate, school records, testing records, etc), which must be done in your city of origin in the US. So while you're at it, make sure you get in touch with the schools to find out all the paperwork they'll need before you come down here. Regardless of whether or not immigration requires one, I would recommend bringing down multiple birth certificates for the entire family. I've found that I've needed them a few times for various things during my time here. Once you're here, they're much harder to get!
I agree with G., better to get birth certificates, certified and apostilled, for the whole family, but most certainly for the children, if they will be enrolling in school, or colleges here. Certified and apostilled records of their studies should also be rounded up before coming down (though at some slight additional inconvenience, this can be done later, if you´ve got someone back home who´s willing to be bothered with helping you get these).
I haven't dealt with INM for my son (he was born here), but we have needed his birth certificate for just about everything else we've ever had to do with him. KNowing INM, the first time you go they will say you don't need it but the second visit they'll ask where it is. I would definitely get an apostilled (or legalized and authenticated if in Canada, no apostilles there) copy and you'll probably need it translated into Spanish too. Just covering all the bases.
Wonder if you can get it apostilled and translated into Spanish, while in the states? Or can the translation be done in Cancun?
The apostilles must be done in the States. The translation can be done in Cancun or in the States. I've done both, and I believe the cost was similar. Keep in mind that the apostilles often have to be translated into Spanish as well.
If your kids will be getting their FM3s as dependents of you they will need BC apostilled and translated (by official translator) and for school they will need at a minimum BC translated. Some ask for apositlle. You will also need transcripts from current schools unless your school allows you (and you decide you want) to enroll outside SEP. I have one kid "inside" SEP and one "outside" at the same school.