Taking pets to Texas

We are moving to Houston in about 2 months with our three cats, and we would like to be a bit more clear about the paper work needed. We will flight from Frankfurt. And we will drive with the cats from Spain to Frankfurt. We know we need the passports, rabbies vacc., deworming,  to travelwithin Europe, but our confusion is what the airline in Frankfurt could ask.

I already contacted the Health department of Texas and they told me the requirements, and there are two things I am confused about and they are

-Internation Health Certificate. They say ,they don't need it, but some airlines could sk for it. I called Lufthansa and they tell me different things!!
-Rabbies Certificate. How is that done? Is it an official paper?

I asked the vets here, and since I am in a little town, they don't have much experience transporting animals abroad, so they are a bit confused also, but they have been very helpful.

Has anyone had moved with their pets from Europe to Texas? And so, tell me their experience?

Thanks in advance.

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