Once we put our Hive account on the Ledger; we can no longer use the Hive Keychain application, or even other connection to interact to write, view, vote, and transfer?
If you say the private keys won't be the same, then the Hive Keychain app where we put our keys will be down.

HBD and HP can be recovered with Ledger, if the Hive platform no longer works or hiveledger.io no longer works?

If you replace all your keys, you will need to sign everything on hiveledger.io or using hive-ledger-cli command line tool. We're working hard to add Ledger support to other wallets as well.