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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 lose control of the creature if the Old Man leaves the battlefield. This is because it is no longer tapped.

The ability doesn’t care whether Old Man of the Sea remains under your control, only that Old Man of the Sea remains tapped and its power remains large enough. If an opponent gains control of Old Man of the Sea, you’ll keep control of the affected creature (until Old Man of the Sea stops being tapped or it no longer has enough power).

Once the ability resolves, it doesn’t care whether the targeted permanent remains a legal target for the ability, only that it’s on the battlefield and its power remains small enough. The effect will continue to apply to it even if it gains shroud, stops being a creature, or would no longer be a legal target for any other reason. If it stops being a creature, its power is considered to be 0.

If Old Man of the Sea stops being tapped before its ability resolves — even if it becomes tapped again right away — you won’t gain control of the targeted creature at all. The same is true if the targeted creature’s power becomes greater than Old Man of the Sea’s power before the ability resolves.