$5,000 to $10,000 is not going to get you very far, especially given that the exchange rate is hovering around $1.40 to the euro. A decent two week trip will cost you about $5000 alone. Also, if your'e going to be staying for 6 months, you're going to need a visa and a residence permit. And place to stay, even like a Wohngemeinschaft (shared aparment), are going to run you somewhere in the range of 300 to 500 euros a month, without utilities. Honestly, I just don't think you can do what you want without significantly more money or a well-paying job over there.
Also, if you're going to learn the language, don't expect to learn it and be fluent in 6 months, no matter how fast a learner you are. German is a very complex language, and can vary greatly by local dialects. "Total immersion" just doesn't work with German; get a good grounding in the basics first.
I'm planning on moving to Germany in the future, and just for the first year, to get established, find a job & a place to live, and improve my German to where I can easily function on a daily basis, I'm planning on needing somewhere around $100,000. And I can already speak a basic level of German.
My advice for you:
1. If you really want to learn the language well, start here with classes to teach you some of the basic grammar & vocabulary. Just being able to read signs & do day-to-day things like grocery shopping will be much easier. Like I said, German is very complex-- there are four cases and three genders (plus plural and formal address), and the articles and word endings change for every one.
2. Do a *lot* more research on living & working in Germany as a foreign national. Try http://www.toytowngermany.com/ for some basic advice. Also, figure out what area you're planning on being in. There's a big difference between cities and towns, and locations. Life in Munich is different from life in Stuttgart which is different from like in Berlin & so on.
3. Save up money wherever you can. Germany is a very expensive place to live. The sales tax/ VAT alone on everything is 19%, plus the horrible exchange rate.
4. Consider waiting a few years. Germany's getting hit by the economic downturn, too. Not quite as bad as the US, but it still hurts. And keep in mind it is law that German citizens get preference in jobs and hiring. An employer who hires a foreigner must show that the person has a skill or qualification that cannot be filled by a native German.
5. Don't give up. Moving to Germany, even short term, is not easy or cheap, but it is possible. That said, I honestly think you need to seriously readjust your plans. Unless you've got a photographic memory, becoming fluent in German in 6 months just isn't possible. You can come for 6 months, but you're going to have a tough time getting a visa and/or residence permit if you don't have a job or enough resources to support yourself. (And to get a residence permit, you will have to prove to the Auslanderamt that you can support yourself-- proof of employment and bank statements are just two of the things you many be required to provide.)
Like I said, what you want to do is prefectly do-able, but not necessarily in the timeframe and within the budget you've stated. Start digging around on the internet-- there are tons of resources out there if you look.