When you go off on your own fork you can do whatever you want, including changing the rules for the block reward subsidy. But again, that only exists on your own fork. It doesn't matter if you start with your own genesis block or extend off of a bitcoin block.
You would want to configure a node to relay transactions received on port 8333 so that they also get relayed to your fork. Some of those transactions will be valid even on your fork but over time more and more transactions that are valid Bitcoin transactions will not confirm on your fork because they include an invalid INPUT as far as your fork is concerned.
The real risk is someone using a copy of their Bitcoin wallet.dat (or the private keys from that wallet.dat) with your fork -- the coins pre-fork are still valid spends on the Bitcoin blockchain as well as on your fork. So it would be way to easy to lose money by accidentally sending valuable pre-fork coins thinking they are only as valuable as the post-fork coins.
Ya, that would get ugly in a hurry.