I'm not exactely 100% on this but from my point of view SHA256 is the backbone of bitcoin.

If you were to take it away and use another it wouldn't be bitcoin anymore you'd just have a new Coin. So yes miners will continue to mine bitcoins because that's the accepted currency.

It would be the same as if another coin was created, essentially ASICs provide insurance that bitcoin is "the" go to coin. Since it has dedicated hardware built for it. ie they can't just flick over to a new coin and start mining that instead.