Closing, please see #249 for further updates.
Your contacts.xml is empty (see https://github.com/rfc2822/davdroid/blob/master/res/xml/contacts.xml)
Consider using my proposed one to enable all fields (like birthday, which is not enabled right now).
Proposed for davdroid: https://gist.github.com/dschuermann/6954672
Also see this example from Google: http://grepcode.com/file/repository.grepcode.com/java/ext/com.google.android/android-apps/4.0.4_r1.2/packages/apps/Contacts/tests/res/xml/test_basic_contacts.xml?av=f
Added contacts.xml and birthday support with commit 6ee010905e172164b865c0e9c0576d92d1ae7906. However, I can’t test it because I’ve only got Android 4.1- devices and they show no birthday field.
I tried syncing a birthday field, and it works as expected.