I just realized that CF7 does not send form emails if used in conjunction with the Cachify plugin - I am also writing about this in the Cachifiy support forum, please contact the author about who has to fix what.
You can reproduce this easily with the stock default form generated by CF7 - it will not send emails as long as you do not put the page id containing the form into cachify exclusion filter.
IMHO CF7 should not be affected by any Cache plugin, please check your JS code, I think there might be an issue if it is reformatted, what never should happen.
What makes me really worry is the fact, that CF7 does not give me any hint about not sending the email. The user is notified about the email not beeing sent, but there is no log, notification or anything about the disfunction to be found anywhere - this must be redesigned to be more robust - you definitely want to get noticed if the contact form on your website does not work. Please add some LOGGING(!) or any other notification support to this plugin ASAP, THANK YOU!
Meanwhile the author of this plugin adds some robustness and advanced error handling to his code, users should use some monitoring service that allows to post forms and gives you feedback about malfunctions.
Yes, we learn this in the first year of software development - if it does not do, what it should, at least it must inform about not beeing able to do, what it should. Of course a notification to the sending user only is not enough.
Please put a BIG notification in the admin dashboard if CF7 was not able to send emails.
Also, please as long as you have not implemented this, put a BIG warning on your CF7 website that explains, that users must setup external monitoring and how to do this, because your plugin is not reliable and has no robust error reporting - if you stay fair now with your users and inform them openly about this problem, they might be not so angry as me when I realized that your plugin just did not send emails without ever noticing me about this error.