Do I need a newer iPad for Capture Pilot, and what screen size works best on set?

I'll precede this question with the caveat that I have no experience with any of the specific tools you mention.

Currently Capture Pilot supports iOS 7.0 and greater. It is certainly possible that they could change this to require a newer version of iOS such as 8.0 and its derivatives. I would say that is somewhat unlikely in the nearterm(year or two) but that is a guess based on my experience with iOS apps. What you want to keep in mind, is that older iPads either do not support iOS 7 or have limited support for it. There is a handy chart here that has an overview of the OS comparability for each iPad model.

At a minimum I would recommend an iPad 3rd generation as that was the first to include the retina screen. Only a few short months after the 3rd generation was announced, the 4th generation came out and has a faster processor that is unlikely to benefit you much for this software but I doubt you will find much difference in price between the two anyways.

Beyond OS compatibility and retina(high resolution) screen abilities, I don't see much reason to upgrade to the latest and greatest iPad model.

Addressing your second and third questions "is bigger really better" and "will it take up too much space?". As I noted, I do not own these products, but I can tell you with quite high confidence that previewing an image on a 9.7" screen with 3x the pixels will be superior to a standard DSLR screen(for my example Canon 5D MkIII vs iPad Air).

In practice, what I have heard is that it is much more typical for the people paying for a studio shoot to be interested in the functionality of Capture Pilot. They want to browse the images as they are being captured on a nice portable large screen, and many photographers are more than comfortable with the screen presented right in front of them on the camera.

If it takes up too much space is up to you really. I don't think that an iPad is very large at all, and I would love to preview studio images on an iPad vs the back of my camera. The same functionality can really be achieved with any laptop or desktop computer though. The iPad just gives you a nice option to hold in your hand vs at a desk, which it sounds like you are interested in.