You can put a gas log set (minus the metal enclosure) into an existing wood burning fireplace. So yes, I believe you could have a standard wood-burning fireplace built and convert it to gas logs. That wood-burning fireplace would have a chimney and I don't know why you wouldn't want to use it. It is an extra expense to build a chimney--see below. Another gas fire chimney option is a double wall flexible tube that runs inside the chimney from the top to the fireplace.
What is shown in your picture is a gas unit designed to be built into standard framing (with clearances and fire resistant sheathing covering the framing). The unit I would chose for that type of build would have a through-the-wall double pipe (combined in one wall hole) vent and air intake. There are a million and one fireplace surrounds you can chose from to dress it up and provide a mantle should you desire one. There are codes to follow regarding the proximity of the vent to windows and doors.
A gas log set inside a double wall metal enclosure--which works as a hot air plenum--provides much more heat when a circulating fan is used. This is pretty much a standard approach for homes that want a bit of flame ambiance, space heat for a room area, and don't mind burning plenty of LP or NG. They are typically vented through an exterior wall which also provides the combustion air from outside.