SHA256 will definitely be broken eventually, stopping Bitcoin mining completely.
Is this really the case? "Definitely?" If bitcoin's algo's become useless to the point of causing bitcoin to completely fail, I'm not sure people would ever have faith in cryptocurrency again. At least not the public. Hell, I wouldn't.
I read time and time again that if catastrophic flaws in SHA256 are discovered, bitcoin is the least of our problems. Not really sure if that is true or not, though.
Every past hash function has failed at some point. It's likely only a matter of time. However, they are usually broken in increments (instead of 256 bits of protection, you only practically get 256 - X bits). Before X reaches 128 bits, I'd expect to see bitcoin (and anything else using SHA256) to move to a newer, more robust algorithm.
Bitcoin won't fail because of this. Even with a partially broken SHA256, difficulty will just go up because miners can use the shortcuts just as easily as attackers. When the time comes, the switch will require a hard fork, but what miner would want to stay with the old, broken algorithm? The biggest threat is if we have two competing algorithms to replace it. However, by the time we get there I would think there would be several companies and individuals in the position (the funds and motivation) to really analyze the options thoroughly and to be able to reach an information based consensus.