It is possible to copy the Bitcoin network and create your own which has no relation with the current one, yes. In other words, you can, indeed, copy the source code, build an implementation of an altcoin client, which has different consensus rules such as total coins, difficulty adjustment period, block interval, genesis block etc.
I've created a thread explaining how:
How to create your own altcoin (v0.20.1)If this is possible would this produce the key to the genesis block?
What do you mean “produce the key to the genesis block”? What's the key to the genesis block?