Your short-stay visa for Germany could take up to 15 days to process. If you send in your visa application during busy times when the German embassies get a lot of them, it could take up to 30 days, or even up to 60 days in rare cases, to process your visa.
All people from EU/EEA countries can come to Germany for up to 90 days in 180 days. They must sign up with the German government if they want to stay longer.
Yes, you can extend your German Schengen visa, but only in rare cases where new facts and special reasons come up after you arrive in Germany. For example, if you must leave Germany for humanitarian reasons or a natural disaster. You should send your extension to the Foreigners Authority (BAMF).
The Schengen visa for short stays is usually good for up to 90 days within six months. On the other hand, the length of time your visa is good for depends on the embassy that gave it to you.
Please send in your application as soon as possible. You could apply up until recently 3 months before your travel date. But under the new rules for Schengen visas, you can now apply for one 6 months before you plan to travel.