Ick at most of these bands. They are not hardcore.
Hardcore is essentially punk with the heavyness of metal, and little more street attitude. Try the Gorilla Biscuits, Minor Threat, Youth of Today, Agnostic Front...that kind of stuff.
You want a more modern idea of hardcore? Try Refused, Jesus Wept, Kids Like Us, Comeback Kid, No Innocent Victim, Madball and that kind of stuff.
You want experimental hardcore? Try Botch, These Arms Are Snakes, the Bled, some Converge, etc.
As for all the something-cores of the world...
Most of those bands have little value. All That Remains may as well be called pop metal, and 'deathcore' is mostly just watered down death metal.
Now as for grindcore, you should definitely look into old Napalm Death and Carcass. Though, older Carcass has quite a bit of death metal even in their earlier stuff(thus creating the original idea of deathgrind, totally seperate from the modern sound of deathcore) it's still got the break-neck speed and chaos of old grind and crust punk.