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

Your "opening hand" is the hand of cards you decide to start the game with after taking any mulligans.

After all players have decided not to take any more mulligans, if the starting player has any Leylines in his or her hand, he or she may put any or all of them onto the battlefield. Then each other player in turn order may do the same.

Leylines that start the game on the battlefield aren't cast as spells. They can't be countered.

After each player has exercised his or her option to put Leylines onto the battlefield, the first turn of the game begins.

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.

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 his or her graveyard at that time.