All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
It seems that it is better not to use this means, i.e. questionnaire, that may give a chance to non-Muslims to insult the Prophet undermine his refined status, and attack his message. Allaah, The Exalted, says (what means): {And do not insult those they invoke other than Allaah, lest they insult Allaah in enmity without knowledge...} [Quran 6:108] If there is a real benefit to be realized through such a questionnaire, then another method should be used to realize that benefit.
It is not accurate to say that in general when someone drives another to verbally declare disbelief, then he himself has committed an act of disbelief. It has been reported that the Messenger of Allaah once passed by children amongst whom there was Ibn Sayyaad. The children ran away, but Ibn Sayyaad kept sitting there and it seemed as if the Messenger of Allaah did not like it (his behaviour) and said to him, "May your nose he besmeared with dust, do you not bear witness that I am the Messenger of Allaah?" Thereupon he said, "No, but you should bear witness that I am the messenger of Allaah..." [Al-Bukhari and Muslim] The question of the Prophet was the reason that Ibn Sayyaad verbally declared disbelief.
Allaah knows best.