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1. The only nuclear-armed torpedo put in service by the U.S. was the Mark 45 torpedo. Its initial operational capability (IOC) was 1963, and design work began sometime in 1960, so it is safe to assume that the first tests of the weapon would have been in 1961 or 1962. There is no official information available about its Soviet counterpart. However, according to Norman Friedman's The Naval Institute Guide to World Naval Weapon Systems (Fifth Edition), Naval Institute Press, Annapolis, MD, 2006, the first known Soviet nuclear torpedo was the T-5 (service designation 53-58) and live tested on 21 September 1955.
2. For the U.S., the first underwater missile was the SUBmarine ROCket, SUBROC, or UUM-44. Work was initiated on SUBROC in 1955 with an IOC of 1962. The Soviet Union came up with a rough analog to SUBROC with the NATO designation SS-N-15 Starfish. According to Wikipedia its Soviet designation was RPK-2 Vyuga, and it had an IOC of 1969.
3. The U.S. never developed an underwater missile that remained underwater for the duration of its mission. The Soviet Union did develop the VA-111 Shkval which is, in essence, a rocket-propelled torpedo. Again, according to Friedman, Shkval's development was authorized in 1960, but was only accepted into service in 1977.
The only part I was fully aware of was the first Soviet nuclear torpedo test in 1955. Here are snippets from a rare (silent) film of the explosion. It will appear in my article on a little-known confrontation at sea between a Soviet submarine and a U.S. aircraft carrier, and later in a book if I can find a publisher.