Who can give me advice about a trip to the USA?

Don't know of a lot of families who take in short-term boarders offhand. There are youth hostels in a few places (check www.hostels.com), allbeit they aren't anywhere near as prevalent as in Europe.

As for where, try Florida - tourism is one of the biggest industies in the state, so the locals tend to be friendly, especially in Tourist Central (better known as Orlando.)

First I would decide on where you want to go. (which depends on what you're into, and when you'll be travelling). then I'd check on craigslist.org rooms to rent or place a want ad for that area.

I don't know of any place in America where people would be hateful just because you're German. There are indeed a lot of prejudiced people in America, but that seems to be more about your skin color rather than nationality. For that short of a stay I'd suggest a large city for the best value and cultural experience - such as New York, Boston, Chicago, San Francisco, Seattle, etc.