just to share solution:
to enable sync while screen is off on android 5.1, you may need to set
Custom battery usage policy for davDroid.
on MIUI it’s located in
Additional settings/Battery & Performance/Manage apps battery usage/Choose apps
find davDroid in list, select
without this - davDroid will fail to connect to server and unable to sync if screen is off.
(tested with 1.0-beta1 version)