You obtain NIE by registering with Foreigners and Borders department of the National Police ("Comisaría General de Extranjería y Fronteras"). See the list of the offices. Note, that since quite a few years, there is separate, much easier and faster process for EU nationals ("comunitarios"), so don't make the mistake of wasting long hours in general queue. As far as I know, since about 2006 or so they don't give residency card ("tarjeta de extranjero") to EU nationals anymore, as formally EU national ID ought to be fully recognized. All you get is a green paper stating that you've registered and your NIE.
BTW. don't get confused, as sometimes people say NIE while actually referring to residency card.
As for Social Security number, usually it's done by your employer. You can also do that yourself by filling in a form TA1 and taking it the nearest office. The page explaining the process mentions national ID, residency card or passport as ID. Note, that you can use NIE as identification number here only if you're using residency card ("tarjeta de extranjero) to identify yourself. As I mentioned earlier, EU nationals are not given that anymore.
So you can do completely independently just using your Dutch passport or Dutch national ID. If I remember correctly, that is what I've done. Downside to that is that they expect the document number to be something permanent, so there IIRC there isn't really an easy way to change that once your passport expires and you get new one.
As I've mentioned in first paragraph, EU nationals don't get these anymore.
Not having Dutch national ID is not a problem, Dutch passport is perfectly fine for all above.
Not having Dutch residency is not a problem either, as you're registering to be resident of Spain. It's not at all uncommon in Spain. There are hundreds of thousands of Argentinians and Brazilians with Italian passports who live in Spain without ever passing by Italy.