Note: This is currently a preliminary answer - perhaps more to come depending on what the OP wants.
Going by the results from the 2020/2021 NHL season (until 7th May) there have been a total of 67 games where the game was 1-0 to either team at the end of the second period.
Of these, there were 6 games where that 1-0 deficit was overturned by the other team ...
The longest ever NHL playoffs streak belongs to the Boston Bruins who made the playoffs from 1967-68 season to 1995-96.
The current longest playoff appearance belongs to the Pittsburgh Penguins who've made the playoffs since 2006-07 season (14 seasons).