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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

You can lose life and take damage, and thereby have a negative life total, while Lich is on the battlefield.

If you have multiple Lich cards on the battlefield, you must sacrifice a permanent for each damage done to you for each Lich. This is because the sacrifice is a triggered ability. But you only draw one card for each life gained regardless of how many Liches you have. This is because the draw is a replacement effect and not a triggered one. You lose if any one of the Liches leaves the battlefield.

If you take more than one damage at a time, sacrifice the permanents for that damage simultaneously. This allows you to sacrifice both a creature and any Aura that is on it all at once.

You can't pay life, just like any player at less than one life can't pay life. You can pay zero life if you want.

If an opponent steals control of Lich, their life total does not change. The life total changes for a player only when it enters under that player's control.

If an opponent steals control of Lich and no other effect prevents you from losing with a life total of zero, you will lose the game due to a zero life total as a State-Based Action before you can take any actions. The last sentence doesn't apply in this case since the Lich didn't leave the battlefield.

The last ability will cause you to lose the game even if you somehow manage to have a life total greater than 0 at the time the Lich leaves the battlefield or if some other effect would prevent you from losing for having 0 life. If, on the other hand, some effect such as that from Platinum Angel says that you can't lose the game then even the last ability of the Lich cannot cause you to do so. The ability will just resolve and the game will continue as normal.

Note that usually you will have 0 or less life at the time Lich leaves the battlefield causing you to lose as a State-Based Action before the last ability can even go on the stack.

The last ability will cause you to lose the game even if you somehow manage to have a life total greater than 0 at the time the Lich is put into the graveyard or if some other effect would prevent you from losing for having 0 life. If, on the other hand, some effect such as that from Platinum Angel says that you can't lose the game then even the last ability of the Lich cannot cause you to do so. The ability will just resolve and the game will continue as normal.