My understanding is that I will need to apply, in Spain, for a residency card after 3 months. I will then be permitted to remain in Europe for the remainder of our trip and freely travel around Europe (assuming the application remains pending or is approved). Is this correct?
This is not quite correct. You are indeed required to apply for this card when staying longer than 3 months. You don't have to apply after 3 months, you can do it immediately after moving to Spain and I would recommend doing that because it can take some time for your application to be processed.
Another subtle nuance is that while the card will make things easier, your right to stay in Spain or elsewhere in the Schengen area is not contingent on it. In fact, the Spanish residence card makes exactly no difference to your status in other Schengen countries. You still have the right to travel freely in Europe (with your wife) or visit any country in the EU for up to 3 months even if you never applied for the card or it got refused.
The thing is that presenting the card is a convenient way to avoid any questions regarding your length of stay or relationship with your wife (e.g. when going in and out of the Schengen area) so it's definitely useful in practice.
Do I need to apply for any type of visa for the initial 90 days? Or can I just travel with my family using just my American passport?
You do not need a visa and can indeed enter just with your American passport. It probably won't even come up but if there are questions about your intent to leave the Schengen area, do point out that your wife is an EU citizen and you are aware of the requirement to apply for a residence card for longer stays.
Will I need to reapply for a residency card in the Netherlands once we move up there for 2 months from Spain?
No you do not and in fact the Netherlands is the one country in the EU where you would not be able to apply for such a residence card (because your wife is Dutch). If you wanted to stay longer than 3 months, things would become more complicated but for two months as a US citizen I wouldn't worry.
To obtain the residency card, we just have to present: (1) my American passport, (2) my wife's Dutch passport, and (3) our marriage certificate. Is this correct?
As far as EU law is concerned, your wife may also be required to prove that she has sufficient resources and health insurance for the whole family or, alternatively, that she is working or studying. On top of that, in practice, I think Spain will require a proof of address of some sort.