1. Sometimes edited content not saved (if you edit in html mode and press save - changed content not saved, you must switch off html mode and then it work).
2. Sometimes I saw several records in falang component to one content and I can edit any of them but result will be the same.
3. Sometimes I can't switch on publishing check, but content has been published.
4. Joomla create single link to home page in any languages and native switcher works correctly with this links (then I click to domain.com, it go to domain.com/lang page), but falang switcher don't work properly and it go to default language page.
5. Custom controls can't be edited in falang
Last update was been several months ago and I don't recommended use this control until author fix this errors.
All your problem are very strange because it's never happens to any user.
Are you sure your joomla installation is well done ?
Look at the other reviews.