We run a very similar dojo model, though I hesitate to call myself a "private" dojo - we use the very similar phrase, "Non-commerical dojo."
We've been running this way since about 1999/00, and we continue to grow and grow. We do have a drop-out rate, but it is almost non-existent.
We don't use a questionnaire but we are interested in this same information - most of it anyways. One way out of expecting students to know the "right" answers to these questions, and/or to get around the almost meaningless value (my opinion) of prior experience, rank, and recommendation, if that is what you want, is to offer a Dues Free Trial Period (in contrast to the regular "One Free Class!"). Ours is a month long (four weeks). In that time, we all get to know each other at a much more informed level. That is an invaluable thing when you are trying to make a decision based upon future commitment. To help this along, we also have a maximum acceptance of five new trial students at a time.
Our place has often been described as very inviting, and community-based, but serious. We've never put off people or rubbed them in a wrong way but for those that were pretending they wanted to train when in fact they didn't really. That said, yes, true, our dojo is not for everyone, though everyone is welcome to step up to the expectations.
We have other such requirements, and we also have other absences (e.g. wholesale mark-ups, federation/membership charges, contracts, ranking exams, etc.), that make the whole thing work together in a way that it is productive and prosperous for all involved and not close-minded, repressive, and/or restrictive.
Here is more information on our membership practices:
I am glad to see what Ross has done here. More folks should try it before they knock it or feel they understand it. Underneath this model is not only a commitment to the art, to one's own practice, but also to one's students. Ross is putting forth a model that should be admired, perhaps even envied.