I am trying to built an application that uses the BIP-32. But I am not sure on how to use it.

I have a passphrase with which I generate the Derived Private Key and Derived Public Key. If I share my Derived Public Key with a second wallet will the wallet be able to generate others Derived Public Key from it ? Is it possible ?

No, it is not possible. The second wallet would need to have either the master public key or the master private key in order to generate other addresses.

And since I have the original Derived Private Key with me, can I see what is happening in the second wallet ?

If you mean can you see the transactions, then yes. This is with any Bitcoin private and public key. If you know one of those, the address can be derived and all transactions can be viewed by anyone.

Also if I loose both the Derived Private Key and Derived Public Key, will I be able to recover my wallets using only the Passphrase ?

Yes. So long as you don't lose the master private key, all of your addresses and the corresponding keys can be derived.