Tenth District Hero

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.