Hi, S.. In my experience traveling with young children is much easier before the age of 1. We were stationed in Iceland from 2004 to 2006, and left the States to move there when our daughter was 3 months old. She did have to have a passport at that time. The tricky part was getting a passport photo of her with her eyes open, as she was only 2 weeks old when we applied for the passports (at that time we had to send the passport photos in with our application package). If things are the same, then yes-your baby needs a passport 100%.
The car seat issue, I am not sure about, but in Iceland they did require them. The cab drivers didn't seem to care if our daughter was in one or not, so if we traveled by cab, (usually just around the base before our car was shipped to us) we did not have her in a carseat; which at that time was an infant carrier/carseat. I am not sure if the 'rules' are different in a different European country, but that is all I know about the carseat question.
The flight could not have been easier with her when she was that small. I was nursing, and I just made sure that she was nursing at take-off and at landing, so that the change in cabin pressure didn't bother or scare her. And then as luck had it, she slept through most of all of the flights, only waking to nurse, which was when there was another take-off or landing. Worked out perfectly.
If your baby is not nursing or taking a bottle anymore, I would suggest a drink that you have to kinda work at to get out of the bottle, like one that you can buy that water goes in with the little pull-up top. Also, I took 2 or 3 NEW toys that she had never seen before, and allowed her to open them on the plane when she was closer to 2. I waited until she was getting bored or cranky and then when I gave her those, she was happy and quiet for hours (the flight from Iceland to Baltimore was 6 hours, and then we had to travel to FL from there). Also, I make sure to have a crazy amount of snacks for her, even now if it is just a road trip, and she is 4 now. I am not sure about rules for taking what liquid your baby would need, but if it is in a bottle, you can buy these little screw-on caps that will fit just about any type of bottle, and then you have a bottle of water or juice that she can suck on to help with her little ears. Here is the website: simplygoodstuff.com
I have heard that the airlines have very strict rules about liquids onboard, but these may help if you have to settle for what they have to sell to you while you are in flight, these will work as long as oyu get bottled water and/or juices.
Also, if you have a seat for your daughter, take her carseat along, and then she will have a comfortable and familar place to rest, and you and your husband will also get a little break if she is able to sleep in it. Plus, you will have it if you find you need it once you get to Spain. If you do not have a seat for her, they can check it as unusual baggage, or something like that, and it is usually fine. We have traveled both ways, and we prefer to have the carseat with us.