Schengen 89 days in each country, 1 EU passport, rest US passports

1. Since one of us is an EU citizen, the whole family does not have to abide by the Schengen 90/180 days rule?

2. Am I correct in assuming we do not have to apply for a residence card if we stay only 89 days in Spain, then cross over to France, and spend another 89 days there, then Germany for 89 days, even though no one gets a passport stamp proving that we went to a different Schengen country?

That's correct, see e.g. Longer EU stay for retired couple

3. Since we spend less than 6 months in a single EU country, we are not tax-residents of any EU country, and only pay US income taxes?

It could be more complicated than that. For example, if France is the country where you stay the longest in a given tax year (without any reference to a specific threshold like 6 months), I think you could technically be considered a resident for tax purposes. But without a fixed address, property or income, I would not be too concerned.

4. At one point would we be required to get EU approved health insurance?

If you stay less than 90 days in any given country, you cannot be required to have insurance. If you stay longer than 90 days, the country may do that but it would depend on national law. There is no EU-wide requirement or “EU approved” health insurance (apart from the travel health insurance requirement for Schengen short-stay visas but that obviously does not apply to you or your family).

5. We homeschool our kids, would staying less than 90 days in Germany create any issues when it comes to Germany having compulsory schooling for children?

No idea. Germany is not the only country with compulsory schooling.

6. Would I have to present my Polish as opposed to American passport when flying into EU, or when renting apartments through airbnb?

I don't think it matters as much as people think it does but presenting it seems more convenient. Whatever you do at the border, be prepared to establish both your citizenship and relationship with your family (this could mean carrying a copy of your marriage certificate) at a later point.

About 5:

Germany:
While each state has it's own regulations, apparently in all of them it only applies to people with the Wohnsitz (ie. permanent residence) there. (One example: http://www.landesrecht-bw.de/jportal/?quelle=jlink&query=SchulG+BW&psml=bsbawueprod.psml&max=true&aiz=true#jlr-SchulGBW1983V40P72).

To make a Wohnsitz more clear: To have that, you need

• a Aufenthaltserlaubnis, which doesn't make much sense for times less than 3 months. (or a Niederlassungserlaubnis or even naturalisation as German citizen, but normally they are not available without having an Aufenthaltserlaubnis)
• to rent or buy something to live
• to register that as your permanent residence
• if it is rented (and not bought), registering needs cooperation of the owner. Hotels won't do this.

Tldr, with less than 3 months in Germany, you don't need to do anything about schools.

...

Like Relaxed said, other countries (including Spain and France) have compulsory schooling too; but you can expect the relevant laws to be reasonable in all countries.