Are these bands metal or not?

As much as people want to deny it both metalcore and nu-metal have played a huge role in shaping metal. Yes some of the bands like Slipknot and Killswitch Engage are hated because of them not being true metal. But unless you fall into the elitist category than you should consider them metal. Its also basef upon interpretation of the category, some view only pure metal as metal. While others are more forgiving and will include both core and other categories in it as well. Both categories have a place in the metal framework. Just how much depends on the person. Me I am not as picky when it comes to this stuff. I just listen to what I want to listen to. But elitism in music needs to stop, its annoying as fu*k. It ruins it for the people who just want to listen.

No. besides Slipknot/Avenged Sevenfold. I would say more like Slayer, Mettilica, Megadeth, Iron Maiden, etc. are metal bands.

Yes absolutely, all of them are, but metal there are a hell of a lot of metal genres out there.

1,2,6: Nope
3: Ersatz borderline metal
4: Closer to metal
5,7: Yes, metal.

They're all under different metal sub genres.
Disturbed and Slipknot are considered alternative/nu metal
BMTH, Killswitch Engage, As i lay dying and Trivium are considered metalcore
Avenged Sevenfold is considered metalcore in their early stuff, after that they did hard rock/heavy metal
But you shouldn't really care about subgenres and if they are metal or not, just listen to what you like ;)

Technically speaking, yes. They're all bands that play metal music.

But are they metal? No.

Those are all Metal bands.

They're more alternative metal or nu metal.

I say yes (well BMTH hasn't been metal since their deathcore days)
Most metalheads would say no because they aren't consider "true" metal (which is a stupid term, just call it metal or put it in a different genre)

I only like Trivium out of those, I don't mind Slipknot.

Oh, and Avenged Sevenfold, as well as, Disturbed aren't metal.