Tenth District Hero

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.

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.

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

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.

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.

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.