At least some of the issues described here seem to be known problems 
with CardDAV, as discussed in bug #12516

The root cause here is that Turba does not expose all address book 
fields via CardDAV, and it also does not delete (on Horde's side) 
values that come back "empty" from the CardDAV client.

Hi Jens, I have an experimental installation with upgraded sync code. 
Would you like to test upon it?

Dear Jan,

I use Horde Webmail 5.2.22 (all updates installed) on Debian 8 and 
Lighty webserver. For contacts sync I use the newest F-Droid Davdroid 
version.

I frequently recognized several issues with contact syncing and I 
don't know if this is a problem with Davdroid, Horde, or maybe my 
webserver configuration.

I can add a new contact on my phone, add an email address or phone 
number. No problems.
But changing a contact leads to the following issues:

1.) A contact has two email addresses (email xx@xx.org, private email 
yy@yy.org). Changing the type of xx@xx.org on my phone to business my 
contact has email xx, private yy and additionally business xx in 
Turba. On the phone the contact has two times business email xx and 
the private email yy.

2.) Deleting an existing phone number in a contact on my phone is not 
performed. After syncing the contact on my phone as well as in Turba 
still has this number.

Deleting or changing contact information in Turba is always correctly 
synced to my phone, so I do all changes in Horde.

Any ideas?

Best,
Jens