Re: The race to the moon started when?
There most certainly was a race to the Moon. At one point the Soviets had three independent Moon programmes underway. They failed because they couldn't get the two best rocket designers to agree on a single design; and when the Politburo did decide on Korolev's N1, they skimped on the funding. Korolev couldn't build big engine chambers to rival those the Americans were designing for the Saturn V, so he used large numbers of smaller engines, making his design complex. Korolev didn't have the ground testing facilities to debug the individual stages of the N1, so it would have to be tested all-up. His tragically early death meant that when the N1 was completed under Mischin, it was a shambles. Four launches, all much later than planned, four failures.
The fallback was to use the Proton - Zond to get a manned capsule based on Soyuz around the Moon before Apollo could do so. The Proton wasn't reliable enough at the beginning and Zond also had serious technical issues - including re entering at crushing G forces and depressurising in flight. A number of Zonds were sent around the Moon, but they only ever carried biological samples.
After Apollo 11 the Soviets claimed to have never been planning a manner lunar mission, saying they could do it all with their Luna sample return missions and Lunokhod. In actuality the manned lunar programme limped on until something like 1975 when the repeated failures of the N1 gave the Soviets a chance to kill their plans in favour of their Salyut space stations and what would become Buran.
Although the N1 is long scrapped, the Soviet manned LK lander still exists and is now on display in Moscow.