Daycare

If you live in Spain longer than 3 months then you have to be registered as a resident anyway, "don't want to do" is not an option.
And yes if you stay longer than 183 days you will automatically become a tax resident and no matter where your income comes from you are required to declare all world wide assets and income to hacienda anyway.
You will also need an NIE as you can't do a great deal of anything without one.

On top of that yes, for schools at least you all need to be registered on the Padron at your local town hall to get a placement. That said you all need to be registered on that anyway if you are living here by law.
You need it for registering your car, buying a car, registering at the Dr's etc...
Also depending on the area you may need the Padron first to register as a resident or vice versa.

You can only get residence in the country you are living in so the Portugal thing wont fly either.
Best stay at home if you want to cheat the system.

I'm not trying to cheat the system at all.. I wont need insurance nor any benefit from the Spanish goverment I have my own!. I can pay for everything from my pocket. I don't need a car, won't be purchasing a house either.  We just want to experience living there and have a change of ambience. We are not free loaders at all!..

Jess    The post about 3 months / 183 days etc. does not apply to you as a non EU national.

  As an American (and all other non EU nationals) you must apply for RESIDENCIA (not the same as EU Citizen Registration).

And the advice about tax on your worldwide income is dubious,  as it is depends on the Taxation Agreement which the USA has with Spain.   There are wide variations.  For example:  The Cananda/Spain agreement does not mean one is taxable in Spain on their worldwide income,