What is the earliest political vote where we have the exact details of the specific numbers of votes and the names of the candidates?
This could be the election of Lucius III.
There were probably 27 cardinals in the Sacred College of Cardinals in 1181. Based on the examination of the subscriptions of the papal bulls in 1181 and the available data about the external missions of the cardinals it is possible to establish that no more than 19 cardinals participated in the election.
As early as 1179, from the 3rd Council of the Lateran, every pope had to be selected from the College of Cardinals, the group casting the votes at the conclave. The number of Cardinals can be different each time, but is known in advance (answering your "specific numbers of votes" question). As low as 7 in under Pope Alexander IV (1254–1261) or Pope John XXI (1276–1277). As high as 1371. Only cardinals under the age of 80 are eligible to vote in a conclave. They are known as the cardinal electors, and their number is limited to 120. The maximum number of eligible electors has been 131 in 20242.
A two-thirds supermajority vote is required to elect the new pope. This requirement has varied since then, depending on whether the winning candidate was allowed to vote for himself, in which cases the required majority was two-thirds plus one vote.3
There are no evidence that the names of the likely candidates only have been effectively recorded. Because of the voting system, each Cardinal can be elected. We can say that there are as many candidates as there are voters, so we know who they are, answering your "names of the candidates" question.
1. wikipedia Maximum number of electors
2. wikipedia College of Cardinals
3. wikipedia Papal conclave