If you stay for three weeks, an apartment would be best, cheaper than a room in a B&B. I assume by family house you mean a B&B, as opposed to a hotel. A gay house? A B&B run by gay owners? I had a quick look at the website for gays in Würzburg, but didn't see any information about accomodation.
Here is the main Würzburg website, I suggest you contact them.
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