I paid 1and1 my payment, but even after showing payment proof they deleted my entire account, domain and all. I only have my Database left (long ago I took a database backup using phpMyAdmin).
Can I get my posts back from it?
Re-install a new copy of wordpress and then put in the credentials to your database in the new install.
If you install wordpress and try to load the domain with no database attached, it will start the site creation wizard. Just enter the correct database credentials (username, database name, password, and location) and it will load for you instead of initializing a database.
If you only have the sql, you'll need to create a new database and import the sql file into this database. the easiest way of doing that is creating the database, then in mysqladmin importing the database. (import in top row of php myadmin once you've selected the database on the left.)
The site won't look good, but when you log into your dashboard, all posts will be there.
From there you can either export the posts (tools/export) or use this database and add a new theme. Photos will not be recovered for pages or posts though