Is this app a wallet? Will we hold our own private / public keys? And can we use an external wallet to hold our Pi in the future?

Yes, your phones will serve as a cryptocurrency wallet that will be linked to your current accounts (number / Facebook).

Like with any other public blockchain, Pi’s blockchain will also allow in-app/external wallets to hold Pi and transact with it, by submitting transactions to the blockchain directly. This is a functionality that we have currently scheduled to be provided on the Phase 3 of the project. (as a reminder we are now on phase 1 – distribution of the currency – we will then launch a Testnet as phase 2, and phase 3 is when our mainnet will be live). During phase 3 of the project, you will also have the option to take full ownership of your private / public keys.
Pi aims to be much more accessible to everyday people which is why we are trying to build this integrated interface. We’ve chosen this phased implementation to make it easier for everyday people to use. We are still in our early days, and are excited for the road ahead of us.