Telegram offers "secret chats". Is it possible that a bot opens a secret chat and sends a message? I could not find any method or parameter for this function.
Telegram's Bot API does not support secret chats at the moment. You can read the documentation, but a short explanation is: end-to-end encryption is used in secret chats which is tricky to implement for bots and requires additional work.
There's no word on when it'll be supported, if ever. At any rate, Telegram Bots are generally meant as a useful aid or for fun, not for sensitive things like financial transactions.