Did symbiote give Peter's Spidey power to venom?

How did venom have spider power like shooting webs and climbing wall? Just like black Spiderman

When it took over Peter, it was using using his powers.

When it left Peter, did it carry over spider powers to Eddie Brock?

Yes. That's also why Spider-Man can't detect Venom with his spider-sense.

In a sense, though I do seem to recall that the whole web being used by Venom was actually part of it's own biomass which mimics the same webbing that Parker's capsules create when shot out. It also did this while It was the suit of Peter Parker.

The Venom symbiote absorbed Spider-Man's powers and conferred them onto everyone who assumed the role of Venom from then on.