Multisite installation problem: new sites with empty names

I’m having big problems getting a multisite WP installation up and running.
I read many posts and isntructions and followed them literally.
I checked mod_rewrite and user privileges.

Although I made some progress, I cannot pass the final hurdle.
Hopefully someone can help pointing me into the right direction.

Here’s what I did and checked:
Fresh install at root level of an unused domain
Created a new database with standard settings (utf8mb4)
Added apache directives to the domain:

<Directory />
  Options All
  AllowOverride All
</Directory>

Ran the WP installer and provided database name, user name and password
added one plugin (jetpack), enabled it and disabled it again
Added the line “define(‘WP_DEBUG’, false);” to my wp-config.php file
Refreshed my browser and follwed the tools-networking link to get the lines to include for the config and htaccess files.
Copied and pasted the config lines above the “/* That’s all, stop editing! Happy blogging. */” line
Copied all lines for the htaccess file, completely substituting what was in there before.
Followed the “log in” link, but before a new log in I cleared the browser cache.

I do see the new “my sites – network admin” tabs.
Adding a site works without error messages, but when I “my sites” shows the sites, only the first site (the first, original one) shows a name. The new one(s) onlyy the WP logo but no site name.
Following that empty link takes me to the first, named site.
When I enter the full path (domain/site/wp_login.php) or follow the link I get by email, I see a database error:
“Error establishing a database connection”

When I check the database, I realize that during the multisite installation new tables have been created (more tables than just after the fresh installation). But they are not populated with additional data; like the new sites.
I can create a new user, which also shows up in the database and gets an email.
Under wp-content there are no new folders. I thought there should be something like sites/ or similar?!?

My mod_rewrite should work. When I add incorrect commands into the htacess I get an 500 error and have to undo the changes so I can login again.

I can delete a plugin from within WordPress, so access to the server is possible.
We run our own dedicated server, so there shouldn’t be an problems from the hoster side.

I have no more ideas what I could do or check.
Do you?!?

Thanks in advance; and fo the patience reading this long summary!
And as final data:
Our server runs apache under debian, at my desk I use Windows 7 and Chrome.
All up-to-date versions.