After I have been accepted to present a paper at a conference, the organisers are asking me to fill in a registration form (participation and attendance are free) AND to produce proof of purchase of flight tickets. I find this bizarre and intrusive. Anyway, my travelling would happen between two capital cities in Europe (plenty of flights) and they demand the proof of purchase to include people in the first draft of the programme. It is a 3-day conference, so I would have to make travel and accommodation arrangements for 5 days many months in advance.
I do not think that the conference organizers have made good decisions about how to handle things. However, it is understandable that they should want firm confirmation that people are actually going to show up. Since they are operating a conference with no fees attached, it is unfortunately too easy for people to sign up, hoping to attend, but then cancel when it turns out the the time or funding are not available. This is a regular issued for conferences and meetings without any direct fees attached, and there are several ways to deal with it.
Some conferences rely on personal relationships with the people who will be participating and presenting. If the people who are going to be giving talks are known acquaintances of the organizers, these connections can be leveraged to ensure that the registered participants will indeed show up. (The day after tomorrow, I am leaving for a free conference, which I am attending even though it conflicts with a family event that I might like to attend. I know the people running the conference, and if I bailed out, it would cost me in terms of my professional relationships. Since I committed to attend many months ago, I am sticking to my commitment.)
The people running this conference are taking a different approach. They are not willing to put people on the presentation schedule unless they can verify that they have made travel plans. While I can understand the motivation for this decision (they do not want people who are slated to give talks to cancel, leaving gaps in the program), I do not think that it is a good approach to take. Asking for verification of flight information seems intrusive and unprofessional. I see no reason to believe that the organizers are acting in bad faith, but I suspect that they may not be very experienced in running academic conferences. So whether you feel like complying with their request for flight information depends on how you feel personally about this unorthodox request.
This seems like an unreasonable request. Given that the conference is many months in the future, you may consider just telling the organizers that you have not purchased plane tickets yet, and will be unable to provide proof of travel arrangements until you actually decide on your mode of travel and precise dates of travel, which will not happen for some time.
While anything is possible, so I do not claim to predict the outcome, all I can say is that the organizers would have to be mildly insane to not allow you to register in such a situation.
Finally, even setting aside the lack of practicality and common sense of the request to provide detailed travel plans so far in advance of the conference date, asking for such details any amount of time in advance of the conference is a violation of privacy and in rather poor form (given that the organizers are not reimbursing your travel expenses, are not your friends, and have no need or right to know your whereabouts at any moment in time other than while you are presenting your paper at their conference). I think it would be completely reasonable for you to refuse to disclose such details on principle, regardless of whether you have already bought the tickets or not.
The organizers are definitely on the wrong foot here. I recommend contacting them and discussing the situation. If they refuse to talk or be reasonable, don't attend. In addition to being "bizarre and intrusive" (to quote the OP), it's also unreasonable for the organizers to dictate your mode of travel. If you wanted to drive to the conference, would you have to prove you own a car?
Although the conference is nominally free, perhaps the organizers should consider the concept of "earnest money" as an incentive for registrants to show up. They could charge some moderately painful amount up front as a registration fee, then refund the fee if the registrant actually attends. No-shows forfeit the fee. This begs the question of what to do with the forfeited money, but there are many options: it will be left, in the best academic tradition, as an exercise for the reader.