My daughter was born abroad, and it takes so much longer to get paperwork back for official paperwork. If I don't have her social security number back soon, can we still claim her on our taxes without it? If so, how?

You must provide a SS number for your dependents. So you should wait to receive your daughter's number before filing your taxes. If you think that will take longer than April 15, file for an extension, which gives you another 6 months time. There's no reason you shouldn't have the SS number by October, right?