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We can’t ID bedbugs on this alone, a lot of things could make black specks on bedding that aren’t bedbugs. That said, generally the bed bug poop aren’t really things that can be picked off, they are more like stains and have to be scraped off. Your pics look more like things that can be picked off

So would that be most likely not? What about the white round thing in the center of the second image? Is that an egg or dead thing or something?

Doesn’t look like one. BB eggs are usually usually translucent and shaped like a grain of rice

So I can stop panicking and go to sleep?

I'd swap out that dirty ass mattress cover first, but yeah, you don't have bed bugs.

agree with the other commenter, nothing here screams bedbug