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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.

If you can't exile enough cards to meet or exceed the required mana value, you can't choose to collect evidence at all.

The effects from Tenth District Hero's abilities overwrite other effects that set power and/or toughness if and only if those effects existed before the ability resolved. They will not overwrite effects that modify power or toughness without setting it (whether from a static ability, counters, or a resolved spell or ability), nor will they overwrite effects that set power and toughness which come into existence after they resolve. Effects that switch the creature's power and toughness are always applied after any other power or toughness changing effects, including these two, regardless of the order in which they are created.

Neither of these abilities have durations. If one of them resolves, it will remain in effect until the game ends, Tenth District Hero leaves the battlefield, or some subsequent effect changes its characteristics, whichever comes first.

You can activate Tenth District Hero's second ability regardless of what creature types it is. The ability checks Tenth District Hero's creature types when it resolves. If Tenth District Hero isn't a Detective at that time, the ability does nothing.

Typically, Tenth District Hero's abilities are activated in the order they appear on the card. However, if Tenth District Hero is a 5/5 legendary Human Detective named Mileva, the Stalwart with vigilance and "Other creatures you control have indestructible" and you activate its first ability again, it will become a 4/4 legendary Human Detective named Mileva, the Stalwart with the same abilities, plus an additional (and almost certainly redundant) instance of vigilance.

Tenth District Hero's second ability checks whether it's a Detective. It doesn't matter how it became a Detective.

Once you've announced that you're casting a spell, players can't take actions until you've finished doing so. Notably, opponents can't try to remove cards from your graveyard to stop you from collecting evidence.