Depending on the cost of the shower unit vs the cost of the cartridge and the time it would take to replace it, we would often recommend replacing the unit instead as this would be more cost effective both for time and money.
anything you drill into you risk causing more damage and of course you must ensure you then have replacement screws to seal the unit properly and fully. otherwise you will get a leak.
That depends what is needed. sometimes they stick and may just need a little lube, or the hole may not have been cut wide enough and it just needs groves cutting for the pets to move smoothly. otherwise, you may be able to just get a replacement button.