Is your concern about the height or the material? How long do the vines get on your specific variety?

IME, climbing vines will climb up anything they can grab onto. If you're worried about height, try inverting a second cage on top of the first one and secure them together. A couple tall stakes inside or outside the cages could provide additional support if it seems like it needs it.

I did the same thing with tomato cages last year and my cucs. I planted burpless and pickling cucs. I did not notice a bit less yield than when I left them on the ground. The plants seemed healthier and I did not have any insect issues (which have been a problem in the past). The vines will climb up and back down the cage then across the ground if you don't keep an eye on them, so I doubt they were any shorter.

My concern is the height of the cage limiting the growth and also the shape of the cage not supporting or promoting proper vine growth.

I'll just see how it does since I'm probably not going to change it now.

The seedling cucumbers in the single pack I bought had 3 tiny stalks all next to each other. Am I supposed to let those all grow or do I pick the healthiest one and allow that one to grow by itself?