How does HD Wallet actually work?

Edit – I’ve refined the question in the post to be more concise.

1. If you derive multiple child/grandchild key pairs from the original master public key, you understand that each of these is a unique, specific address. But if grandchild key pair 1 contains 1 BTC and grandchild key pair 2 contains 2 BTC, can I send a total of 3 BTC in a single transaction? If so, will both appear on-chain to escape the derived subkey pair address?

2. If the master private key is in cold storage, but one of the child private keys is not in cold storage, can that wallet be defined as a cold wallet?

thank you