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JotForm is a free online form builder which helps you create online forms without writing a single line of code. No sign-up required.
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Adding the form to facebook page is working fine when I tried it just now.
Here's the guide: https://www.jotform.com/help/22-Adding-a-form-to-your-Facebook-page
Here's the video tutorial:
Make sure the pop-up option isn't blocked in your browser, otherwise the login window will not be shown.
I talked about the "Login with facebook" option which was suppose to achieve by integration. However I tried all mentioned way I found here. Still couldn't figure out. See screen shoot for detail.
First off, we suggest to avoid creating multiple threads. This will only create confusion.
With regards to your main question, this was already answered and escalated to our developers by my colleague on this thread https://www.jotform.com/answers/869222.
If you have follow up questions, feel free to post it on that thread.