No, it is not mandatory for green card holders to posses a machine readable passport (as the green card is machine readable).
No - a Green card means your a permanent legal resident. ESTA is only used for Visa holders and waivers
Yes, green card holders need a visa to travel to Chile.
Yes, green card holders need a visa to travel to Thailand.
Yes, green card holders need a visa to travel to Italy.
Yes, green card holders need a visa to travel to Japan.
Yes, green card holders need a visa to travel to the UK.
Yes, green card holders are allowed to donate to political campaigns in the United States.
In most cases, a country's immigration laws only care about your country of citizenship. Having a green card gives you legal permanent residency in the United States, but does not grant you any special status to enter other countries. You will have to determine what Panama's laws are regarding your country of citizenship, which is the country that issued the passport that you will use to travel.
Yes, green card holders are required to obtain a Chile visa for entry into the country.
Yes, they can.
Yes, UK transit visas are required for green card holders traveling through the UK.