I am growing courgettes in 10-gallon nursery containers for trees, I grew 3 plants per pot. My courgettes are nearly done fruiting (note that my geography is the State of Georgia, in the United States, that makes a difference). I will probably sow seeds for new plants in early-mid August for a second crop.
I would not trim any leaves except leaves that are yellowing due to age or disease. The leaves are the 'solar panels' that feed your plant. The stems and flowers need the sun far less than the leaves!
I see mostly male flowers in your pictures (long stems, usually start growing on the plants first). Female flowers start appearing later, and the base of the flower is a small version of the fruit (which will fill out if the flower is pollinated), on a short stem. Is it possible that you don't have any female flowers yet? That would explain the lack of fruit.
Your plants are young and still small, so do allow some time for them to grow. Courgettes are heavy feeders, so I would keep up with providing regular soil nutrition to your plants. I provide liquid feed at the recommended strength, it is a balanced N-P-K fertilizer. Other members here might recommend a different N-P-K balance, nothing wrong with that. I provide that every other week, more after prolonged heavy rain events.
You can stake the plants, but you can also let the plants sprawl out of the pot onto the ground. That is what I do.
