U.S. retirement benefits normally need 40 quarters (10 years) of credits. If you split a career between the U.S. and Slovak 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 Slovak 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. Slovak Republic's authority issues the equivalent when Slovak Republic covers you.
Under the U.S.–Slovak 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 Slovak Republic's system. Your employer obtains a U.S. Certificate of Coverage as proof. Beyond 5 years, coverage shifts to Slovak 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 Slovak 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 2014.
Pension taxation depends on Slovak Republic's domestic law and the separate U.S.–Slovak 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.