is black doom dead or alive?

Absolutely. Shadow killed that (explicit) in "Shadow the Hedgehog!

Most likely, he's dead by the end of his boss fight. HOWEVER, judging by how resilient Shadow is* and how Shadow was made from some of Black Doom's own DNA**, it's a possibility, even if it's a small one -since Shadow kicked his *** while in a super form- he could still be alive after their fight.

*He survived falling from space, and even with robots sent to save him (Eggsy was talking to Tails about his Grandpa /after/ we see Shadow flicker out of existence), he should be dead, like, really dead. (Which he was meant to be, but his fanbase saved 'em.)

**Hedgehogs could also be invincible, haven't you seen Sonic be impaled by spikes and he keeps going fast?

Dead. He was destroyed by Super Shadow.

he kill him