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There should be shut offs under the sink. If you don't have any all you can do is put a rag over where it's leaking to knock the flow down.
If you have more than 2 shut offs then they could go to the shower. Another possibility is integral shut offs where the water lines tie into the shower valve in the wall. You'll have to remove the handle first then the trim plate at the wall. Behind the plate if you see stem with slots where the water lines connect to the valve (the pipes into each side of the valve)those are shut offs and you can turn them with a screwdriver to shut it off.
I'm afraid not. The shut offs right below the fixtures are for just that fixture. Your best bet is to remove the trim plate at the wall behind the handle on the shower faucet.
The screws should be brass. They shouldn't rust. I don't see how you can get to them with a hack saw. You could use a drill to drill the heads off but be careful to use a it that's smaller than the head. Nothing bigger than a 1/4" bit.
You shouldn't hit any water lines by drilling the heads off, If you can slip a hacksaw blade behind the plate parallel to the wall that would be OK.
Put a small towel over the handle. It will knock the water stream down
You have to take the handle off anyway to get the plate off.
You're only option here is to drill the screws off if they are rusty. A hacksaw won't work. Some moron used steel instead of brass screws if they are rusty. But given the likelihood you'll get the drill wet drilling them off isn't a good idea.