When is the nonce generated? What parts of a transaction can be changed after the nonce was generated? How is the nonce connected to a transaction's children/parents?
The last step, after bundle and tips have been set up, is calculating a nonce matching the minimum weight magnitude (MWM). On the mainnet, MWM is 14, which means the transaction hash has to have 14 trailing 0-trits.
None. Transaction hash is calculated by absorbing all transaction trytes and create a 81 characters transaction hash. This means, if you change any characters, it will give you a complete different transaction hash.
It is not connected. Nonce only affects that transaction (not the bundle, and not other transaction in the bundle).
For how a transaction were generated in IOTA, you can take a look at this article: In-depth explanation of how IOTA making a transaction (with picture)