I have a Polish and American passport, but I have never lived in either of those countries, I have instead lived in 6 Asian countries and now the Netherlands. So I am an EU citizen but… I have hardly any connection to Poland, technically my home country? Lets move onto my question, which is actually kind of two fold. Firstly, could I buy a car in Poland, pay Polish insurance, and Polish road tax while living in the Netherlands? If so, how long could I do this before having to drive it back now and then. On the other hand, can I do this with a car from any EU country?
Means of transportation are excluded from many rules regarding the EU single market and EU countries, including the Netherlands, often impose specific taxes on cars. To prevent circumventing these rules, many countries impose severe restrictions on using cars registered elsewhere than in your country of residence (citizenship is typically irrelevant and non-residents like tourists are fine using their EU-registered car too).
There are exceptions if you just moved or are a student and a bunch of court cases about company cars and the like but in general, as a resident, you are not allowed to drive a car registered abroad without paying for the Dutch vehicle tax (bpm) and yearly road tax (mrb), even for a single day. If you need to use a car registered abroad for less than 14 days, you can apply for an exemption online. You get only a single 14-day exemption per year for a given number plate, which tells you how restrictive the rules are (this is intended for cases like returning home with a rental car after an accident abroad).
Because owning and registering a car in the Netherlands is quite expensive, even compared to other EU countries, I know several people who have done what you envision and pulled it off for a period of time but the law forbids it and is actively enforced.
Firstly, could I buy a car in Poland, pay Polish insurance, and Polish road tax while living in the Netherlands?
No. You can buy the car in Poland, but you will need some insurance to drive it to the Nederlands, but you will have to register it again in the Netherlands.
The issue of the insurance may be more tricky, I do know of some people in Spain using a UK insurer but I do not know if it would be applicable to any EU insurer, or if there are just some that offer this service.
And of course, maybe the Polish insurers may object to covering you if you reside at the Netherlands (with presumably more expensive garage shops); check the insurance policy.
If so, how long could I do this before having to drive it back now and then.
It does not work like "visa runs". The "counter" does not reset just because you move the vehicle out of the Netherlands. It is your status as a resident that matters.
On the other hand, can I do this with a car from any EU country?
The same rules apply for any EU car.
The EU common markets leaves outside some excise taxes (as it does for tobacco and alcohol duties) that are to be paid to each country. If you become a resident of the Netherlands, you can bring your car but that will be an import; you will have to register the vehicle in the Netherlands and pay whatever taxes the Dutch government decides to apply.
This pdf explains that you qualify as a Dutch resident if you have lived there for more than 4 months during the last 6 months. At that time you no longer are a visitor and you must register your car.