Hey, I’m a developer myself and I know how annoying it can be when people say vague things like “doesn’t work”. Rest assured that it’s nothing simple or obvious, as I went through all those things myself for eight hours, including trying to filter something useful out of the DAVdroid and apache logs and even analyzing the network packets, before I gave up and posted here.
The facts are these:
• I tried it with 0.6, and even though KDE/Akonadi/Kontact and Thunderbird/Lightning/SOGo worked flawlessly, DAVdroid did not. I checked for typos, any kind of connection problem, tried every thinkable path, mod_rewrite, apache config, DAViCal config and security grant/setting.
• Somebody on IRC said the exact same version of both DAVdroid and DAViCal worked for him. But when now creating a new sync, he got the same “Invalid CalDAV response” error that I do. So I asked him if he had originally configured it back before 0.6 was out, and he said he did… around March or April.
• So I went through the old versions of DAVdroid available in F-Droid. And while all the older ones failed due to being unable to support TLS 1.2 or similar TLS problems, I finally got to create the syncs, and it to function perfectly with 0.5.14-alpha.
• The person I spoke to on IRC confirmed that it kept working when going back to 0.6, provided one keeps the old sync config. So the problem only occurs when creating new syncs.
All this makes it really clear that it’s a kind of bug in 0.6.
I will try 0.6.1 or later as soon as it’s available though F-Droid. (I’m rather poor right now. Sorry. All I can offer is a place to sleep or where to find nice things in my city that no tour guide will ever cover. 
But it’s definitely not a password problem.