Are there some problems with svn.kde.org? I cannot connect to checkout the sources to translate. I have been translating KDE for a few years, but a few days ago it stopped working. Does it mean that we should move to git?

Nope. If you are the holder of a KDE commit account then you should have received a mail from kde-cvs-announce a few days ago, concerning changes to Subversion access in KDE.

If you don't have the mail, then I can post the instructions here as well.

I am having problems connecting to svn.kde.org. I am getting these errors:

Permission denied (publickey).
svn: E170013: Unable to connect to a repository at URL 'svn+ssh://svn@scm.kde.org/home/kde/branches/stable/l10n-kde4/sk/messages'
svn: E210002: To better debug SSH connection problems, remove the -q option from 'ssh' in the [tunnels] section of your Subversion configuration file.
svn: E210002: Network connection closed unexpectedly

What does it mean? I just upgraded from Kubuntu Wily to Xenian and my id_dsa file stayed the same, also KdeSVN aplication and all other applications and config files.

Debian (among others) disabled support for DSA SSH keys as part of a security update.
You'll need to re-enable this (I believe a single line in your ~/.ssh/config is all that is required) or generate a new key.