Yes.

Yes, but if you pay taxes in the country you're staying in, there's a chance you can deduct that tax from your taxable income on your US taxes, depending on the treaties in play.

yes.

Not necessarily. A number of countries are exempt from double taxation.

The United States has tax treaties with a number of foreign countries...Under these same treaties, residents or citizens of the United States are taxed at a reduced rate, or are exempt from foreign taxes, on certain items of income they receive from sources within foreign countries.

https://www.irs.gov/businesses/international-businesses/united-states-income-tax-treaties-a-to-z

You are still required to file, however, and usually you are able to use the Foreign Earned Income Exclusion to avoid taxation of most of your income as long as it doesn't exceed a certain amount.

If you are a U.S. citizen or a resident alien of the United States and you live abroad, you are taxed on your worldwide income. However, you may qualify to exclude your foreign earnings from income up to an amount that is adjusted annually for inflation ($107,600 for 2020, $108,700 for 2021, $112,000 for 2022, and $120,000 for 2023).

https://www.irs.gov/individuals/international-taxpayers/foreign-earned-income-exclusion