I am having the same issue with DavMail. It seems that the DavMail Calendar shows up then gets deleted in some sort of loop. One min the calendar show up the next min it's gone. Then it's rinse and repeat.
The calendar is fine pre ios7 I have an iphone 3 where it is still working fine.
Would love to see this fixed.
Actually, this is a User-Agent problem.
Davmail does not recognise the user-agent of IOS 7 which is causing the problem.
The latest CVS of davmail has the fix, and I can confirm that it works (even though it was jumping through hoops to get it to compile on my rather messed up CENTOS 5 system).
I would guess that the next release version of davmail will have this fix included.
Otherwise if you can't wait, go for this:
Use CVS to chekout the latest version (As per instructions above) - I built 4.3.4-2184-trunk
I had the very devil getting the build to accept that I had Java 1.7 installed, and had to modify the build.xml file to remove the java version check, and all references to it, and them look for the compile target and modify the java 1.5 to java 1.7 (both source and target)
I tried build 2187 of DavMail and it is better, but no cigar. The calendar is not disapearing from the list of calendars anymore, but I don't see my appointments. It's interesting to note that the Mail app in OS X Maverick is working fine. Also, still have the MIME issue on IOS 7 with build 2187.