Adding a supertype doesn't overwrite other supertypes. Each nonland permanent will be legendary in addition to its other supertypes.

If the game starts with more than one Leyline of Singularity under a single player's control, all but one of them will be put into their owners' graveyards right after the upkeep of the first turn starts. If the game started with multiple copies of any other Leylines under a single player's control, that player will also put all but one of them into their graveyard at that time.

The "legend rule" says that if a player controls two or more legendary permanents with the same name, that player chooses one of them and the rest are put into their owners' graveyards. Notably, if you control Leyline of Singularity and two or more creature tokens with the same name, only one of the tokens will be allowed to remain on the battlefield.

A player's "opening hand" is the hand of cards the player has after all players have taken mulligans. If players have any cards in hand that allow actions to be taken with them from a player's opening hand, the starting player takes all such actions first in any order, followed by each other player in turn order. Then the first turn begins.