i've been trying to get davdroid to run with calypso. apart from having to introduce autodiscovery in calypso (don't give in easily to those who demand manual configuration, servers should just work ), i had to patch my way server-side around davdroid appending slashes to urls even if not specified. in particular, when the server uri as entered in the dialog or the principal url don't end with a slash, they are requested with a slash anyway.
usually, it's no good to mess with slashes at the end of uris. i'm not very familiar with webdav, though. if davdroid's behavior is to spec, please point me to the specs; otherwise, please don't append slashes.
Collections ("directories") have a trailing slash at the end, while non-collection members don't. See RFC 4918 page 15:
"Wherever a server produces a URL referring to a collection, the server SHOULD include the trailing slash. In general, clients SHOULD use the trailing slash form of collection names. If clients do not use the trailing slash form the client needs to be prepared to see a redirect response."
WebDAV/CalDAV/CardDAV are quite complex protocols and many servers return invalid responses (for instance, unencoded=invalid URLs that make parsing impossible) etc. So I suggest that if you implement these standards that you try your best to be compliant, at least for the important features.