I use them all the time... I have a site set up using 1&1 on a shared server right now. It is a problem with your site settings and not a problem with our recommendation. Believe me... I'm sure on this one.
Dreamweaver posts a file to your testing server and posts to it for all database interaction... this means you don't need to access it remotely. It works with a direct connection on your hosting server. You are barking up the wrong tree on this one. The issue is not with 1&1 hosting... your issue is with your site sittings.
I just set up a site on 1&1 shared hosting yesterday and have no issues connecting to the database.