So basically, the maximum 21M supply will never be changed when the first block was mined.
Correct.
You can. As long as you modify the code to increase the block reward and people run it, you would have effectively increased the total possible coin in circulation. The change can only be for those running your modified client.
It wouldn't be considered Bitcoin anymore though, since the coin cap violates one of the few goals that Bitcoin is trying to achieve.
You can but no one will take those modified and increased supply coin, just like those forked coins.