will this EVM integration allow us to swap our HIVE for BTC, ETH, or USDC and vice versa directly within your ecosystem?

Yes, all this directly through your Hive wallet.

I would also like to know if I will be able to store and manage those coins from my own Hive account using interfaces like PeakD, or if using an external wallet like MetaMask will be mandatory.

You can already go to altera.magi.eco and connect your Keychain to Altera and hold Bitcoin inside your Hive wallet just like you can hold Hive inside a Metamask or a Xverse (btc) wallet.

Any blockchain Magi integrates with allows you to hold those tokens inside your Hive wallet.

I would also like to know if I will be able to store and manage those coins from my own Hive account using interfaces like PeakD.

You can manage it via https://altera.magi.eco/. And yes there will be options via Keychain or Peakd as well.

Lastly, how do ZK-proofs help keep my funds safe when moving them between networks?

ZK proofs mathematically guarantee that every deposit and withdrawal is backed by a real, verified transaction on the source chain, in this case Ethereum.