Where does the cooling unit exhaust under a staircase?
Through-wall heat can go into the big living area beside the stair if that room is air-conditioned and ventilated. A coat closet or cupboard next to it is ruled out, so the alternative is ducting the heat away or running a split system to an outdoor condenser.
Does an under-stair cellar need cooling?
For storage beyond a few months, yes; the space follows the house temperature, which in a Texas home stays above the 50–60°F range recommended for long-term storage.
Can it be glass?
It can. An insulated double-glazed front and a glass door sealed on every edge put the bottles on show, and the cooling is sized up for the extra heat that glass admits compared with a framed wall.