U.S. retirement benefits normally need 40 quarters (10 years) of credits. If you split a career between the U.S. and Czech Republic, you might fall short in each country alone. As long as you have at least 6 quarters of U.S. credits, the agreement lets you add your Czech Republic credits to reach eligibility — the U.S. then pays a benefit pro-rated to your U.S. credits only.
A Certificate of Coverage is the document that proves you're exempt from one country's social security. For U.S. coverage, the employer requests it from the SSA Office of Earnings & International Operations. Czech Republic's authority issues the equivalent when Czech Republic covers you.
Under the U.S.–Czech Republic agreement, a worker sent by a U.S. employer for 5 years or less normally stays under U.S. Social Security only and is exempt from Czech Republic's system. Your employer obtains a U.S. Certificate of Coverage as proof. Beyond 5 years, coverage shifts to Czech Republic.
Yes. If you have at least 6 quarters (about 1.5 years) of U.S. credits, the agreement lets you combine ("totalize") your U.S. and Czech Republic credits to reach the 40-quarter (10-year) threshold for a pro-rated U.S. retirement benefit.
The agreement has been in force since 2009.
Pension taxation depends on Czech Republic's domestic law and the separate U.S.–Czech Republic income tax treaty (if any), not the totalization agreement. The totalization agreement only governs which country's social security system covers you and how credits combine. Confirm pension taxation with a cross-border tax advisor.