Not exactly. It's individual health insurance from the federal government. If you're a U.S. citizen and you've been working, you've been paying for Medicare all along.
No. A lot of people confuse Medicaid – health insurance for people whose income falls below the federal poverty level – with Medicare. You don't qualify for Medicare based on income. Medicare covers seniors ages 65 and older, and some others who have disabilities.
Exactly. Most people don't have to purchase Medicare Part A (hospitalization) because they've been "buying" it through their payroll deductions for years. However, if you want Medicare Part B, you start paying for it when you become eligible for Medicare.