I married into a Spanish family and spent 4 months in Barcelona in college.
1. Does this sound OK? (I am obviously leaving out Madrid but I am finding it very hard to find anything interesting besides the palace)
It sound OK if it's what you want to do. The palace in Madrid wasn't something I found all that interesting, but I loved the Prado (top tier art museum) and the Reina Sofia (museum housing the 20th century pieces that used to be part of the Prado collection). Also various parks and plazas, restaurants, etc.
2. Am I missing anything?
Spain is a widely-varied country. You're planning to see a part of Catalonia and three cities in Andalusia. You're missing most of the coast and all of the Basque Country, Galicia, central Spain, etc. That said, there's only so much you can do. If you have your heart set on Granada/Sevilla/Cordoba, more power to you, but I found the food, sights and architecture of those cities to be very similar.
3. I want to watch a Barcelona game at the Camp Nou.
During the season, football teams in Spain typically play one game per week, alternating home and away, and virtually always on a Saturday or Sunday. Games are also usually late at night, which may not work well with a 4 y/o. (In general, Spain has a very different definition of "late at night" than the US. Most restaurants won't open for dinner until 9 and eating dinner at 9 is about equivalent to eating dinner at 4:30 here. Between 10-11 is most typical.) So, if your schedule works out that you will be in Barcelona while FCB is playing a home game, go for it, but understand that it's not likely unless you are solely planning around the game. That said, Camp Nou has a museum and offers tours even when the team is not playing.
4. We will be traveling with our 4 YO.
Great! I've found that Spaniards are much more accepting of children in what would be "adult" spaces in the US (restaurants, coffee shops, museums, etc.), but there also tends to be a higher expectation of their behavior. If your child misbehaves, the expectation is that you will step out with them until they are ready to return; there isn't so much of a "kids will be kids" attitude. When I was last there, restaurants would often bring my son (1.5 at the time) treats from the kitchen (one time it was a giant plate of house-made potato chips) and pay special attention to him.
5. We are vegetarians. What do we do?
First, understand that just because something looks vegetarian, it may not be. Also, standards of what is and is not vegetarian are completely different in Spain. Chicken and other poultry aren't considered meat, nor is lard, and the concept of veganism in most places is met with utter confusion. This is going to be most true in Andalusia, less so in Barcelona.
Historically, Andalusia was Al-Andalus, the heart of Muslim Spain. After the reconquista, many Jews and Muslims converted to Christianity and it became commonplace to add pork products to virtually every dish in order to prove your true Christianity. This means that many, many traditional dishes looks vegetarian (because they're peasant food and peasants couldn't afford much actual cuts of meat) but contain lard or other pork products.
In Barcelona, you're going to see much more seafood, and seafood products or fish stock in a lot of dishes. A vegetable paella, for example, will usually have fish stock and clam sauce.
What are your particular restrictions? If you're OK with eggs, tortilla española will almost always be OK to eat (and ubiquitous). Otherwise, do some reading on traditional dishes and their usual ingredients and learn how to ask about concerning ingredients. (For example: Is the vegetable paella made with fish stock? ¿La paella de verduras está hecha con caldo de pescado?) Simply asking if a dish is vegetarian isn't likely to get you the right result.
6. Do we need a car? With a 4 YO?
No, and I wouldn't recommend it. Public transit, both within and between cities in Spain is excellent.
7. How soon should we make train (renfe) reservations?
The earlier the better, in order to get the cheapest tickets. If memory serves, Renfe tickets are available a year in advance, so you should be good to book anytime.
8. Where to stay?
I have had good luck with private rooms in hostels (free breakfast and it's easier with kids to have a kitchen, I think), pensions and Airbnb apartments. In Barcelona, I'd recommend staying in the city center. Barri Gotic is the old city, which is beautiful and medieval, but kind of expensive. El Ravel and La Ribera are the hip, artsy neighborhoods flanking it. Les Corts is where Camp Nou is located; it's a Metro ride from anything else but is a great local upper-class neighborhood.
A few other notes:
-Be aware that while Spanish is spoken and understood in Barcelona, it is not the local language. That is Catalan. It's helpful to be able to read a few key phrases as most signs, if they're in only one language, will be in Catalan.
-There isn't really an "off season" for tourism in Spain. Be prepared for crowds.
-Barcelona has a problem with pickpockets; they primarily target tourists. Do whatever you can to not stand out, and don't be obviously inebriated or lost. If you do get pickpocketed, the best thing you can do is confront them in Spanish. They are looking for easy prey and will often give items back once they realize that you're not that.
