So since it is a rental place, do you want to or can you put holes in the wall to mount a swinging gate? Do you have room to open a gate that wide? Have you looked at commercially made expanding baby gates? People don't need baby gates for very long and I bet you could find a pile of them on Craigslist. I'm not trying to talk you out of building one but I wonder if you might find one that would work for your current location.
If you're set on building one, the gate should be pretty straightforward. Simple mortise and tenon joinery--maybe bridle joints at the end for looks. Use maple or cherry and round over all the edges and sand them smooth so no splinters orsharp corners for the little cherub to get damaged on. Gate or T-strap hinges on one side and either a sliding bolt or cooler, a spring latch made of maple.