Can you recommend a shower that has the same color?

Yes, please have a look at this link: https://www.homary.com/item/chester-rainfall-showerhead-with-handheld-shower-exposed-shower-system-antique-black-11388.html

Do you have a pop-up drain to match this sink faucet and color?

Yes, you can have a look at this link: https://www.homary.com/item/modern-solid-brass-pop-up-sink-drain-with-overflow-for-bathroom-sink-in-matte-black-10637.html

Can this unit be used in a three-hole opening?

Yes, it can be used in a three-hole opening.

Does this include the deck plate?

No, this item does not include the deck plate.

Can you recommend a tub faucet that has the same color?

Yes, please have a look at this link: https://www.homary.com/item/single-handle-black-geometric-bathroom-sink-faucet-single-hole-solid-brass-architectural-design-15302.html