1. Yes they did! The "Kempei Tai"(sp?), feared throughout Asia.
2. The Russian detonated their first bomb on the 29th of August, 1949, almost exactly 4 years later after the US.

And they BUILT their first nuclear weapon using Nazi engineers and stolen intelligence from the US.
3. As Kennelly stated, on the 4th of October, 1957, Sputnik I became the first man-made object in space, orbiting Earth in 90 minutes. Transmitting on 20 and 40 MHz the characteristic beeps which scared the "sh*t" out of the Americans!
4. Well, the Space Race was long and so full of great achievements, I don´t know where to start, be more specific.
Russians:
First object in space (Sputnik I), 1957
First dog(animal) in space (Laika), 1957
First probe landing on the moon (Lunik I), 1959
First human in space (Juri Gagarin), 1961
First probe on Venus (Venera 3), 1965
First probe on Mars (Mars 2), 1971
First space lab
First space capsule connection
First space station
Even bigger and better Space Shuttle (Buran), not started though
US:
First manned Moon landing (Apollo 11), 1969
First probe to leave solar system (Pioneer 10), started 1971, left 1986
Farthest away probe (Voyager 2), 1977
First Space Shuttle, 1981
That´s what I can answer to these questions from memory, not enough time for more.
