#578 is caused by https://github.com/owncloud/contacts/issues/581 which is present since at least OwnCloud 6.0.3.
Please follow up in the issues mentioned above.
Please support the use of self signed certificates without required workarounds. Other apps like CalDAV Sync (https://play.google.com/store/apps/details?id=org.dmfs.caldav.lib) show an advice the user have to confirm, before doing the sync, so no workarounds requred.
This issue has been raised before. See https://github.com/bitfireAT/davdroid/issues/3. The short answer is that you can use CAdroid to import your own certificates.
In point of fact, if DAVdroid were to support self-signed certificates, DAVdroid itself would be doing a workaround of Android’s certificate management process.
Duplicate of #3, please follow up there.
I have recently seen a certificate without CA flag that could be imported and used with DAVdroid in Android 4.4, so when I find time, there might be a CAdroid update that allows certificates without CA flag to be imported in certain Android versions. That would greatly improve user experience for DAVdroid users, too.
For more information, please see in #3, https://davdroid.bitfire.at/faq/entry/importing-a-certificate, and http://cadroid.bitfire.at/.