How can I transfer or link my Clash of Clans account to an Android device, from iOS?

How do I retrieve my Game Center username and password from my iPhone to my Android, so I can restore the progress of my Clash of Clans app?

You don't need your Game Center user and password —which in any case is usually your Apple ID and password; you can link devices in-game.

From the Supercell support site, specifically "Transferring your game to another device":

iOS > Android or Vice Versa

First, make sure the game is installed on both devices, keeping them both on-hand. Use the “Link a device” feature in the in-game settings, and then select “Link a device” on both. Follow the on-screen instructions to complete the transfer.