Can I buy a SIM card in Germany for a short duration?

Try lebara.de, who offers SIM cards specifically targeted towards people who want to frequently call internationally. You can find one at almost any kiosk. They advertise rates to India of only 1 cent per minute (plus a 15 cent per call connection fee). I don't know what the catch is.

Another option might be LycaMobile. They advertise free or very cheap rates when calling other LycaMobile customers, even internationally.

Ortel is another provider advertising international calling from a mobile phone for 1-3 cents per minute.

I only know of these guys because I saw their ad in a kiosk window earlier today, just to illustrate how easy it is to get a SIM card in Germany:

The German Telecommunication law requires carriers/resellers to register the name, address and birth date of every end user. Some will only accept German addresses or additionally require a passport number.

And yes, calls to India will be very expensive unless you get an additional callthrough card. I've also read that some carriers block such callthrough numbers - T-Mobile and Vodafone (and resellers in those networks) supposedly don't.

The law to register your SIM card with an ID or passport is stricter now, which means that you need to identify yourself at the post office (POST IDENT) or online. Online is pretty easy and you can do it within 10 minutes. When you order one of these prepaid cards from O2 or Vodafone, they will already ask you to provide your correct address (as in your ID or passport), because it will be checked afterwards.