Is this a long or short term move?
Unless you're planning on making this a very long term move, you may want to skip sending over your furniture if you want to keep costs down. If you go back to the US, you may need to pay for your belongings to be returned (if possible, negotiate this in your contract, but keep in mind you may not be working for the company by the time you'd return, or they'd only pay if you return within a specific amount of time such as 1 or 2 years). Don't ship over furniture that you love and would want to take back to the US unless you're coming for a period of more than 4 or 5 years.
Keep in mind, also, that room sizes (and flats/houses overall) are smaller than in the US, and bed sizes are different, so finding bedding for your American mattress will be more difficult.
Flats here frequently come furnished, so you'd have that to consider as well, when bringing your things along.
That all being said, I'd think it was doable. But I don't think they should be charging you for getting you and the family a visa, that should be a business expense and not related to your personal relocation expenses.