They also are legally required to collect donor information for donations above $1,000 [1]. Mostly this is to ensure that nonprofits aren't playing weird games with funneling money around, e.g. being registered as a 501(c)(3) charity but actually being funded mainly by one person who also controls the board/spending.

Maybe they can rework the donation flow so it allows anonymous donations below that limit? But if that has the effect of making large donations seem different/weird/complicated, it might be counterproductive: you have to get a lot of extra $5 donations to make up for a single lost $10,000 donation. I'm no UI/flow expert though so maybe there's a good way of doing it. There are basically three cases that would need to be worked in: 1) above $1,000, information is mandatory; 2) between $250 and $1,000, information is not mandatory but must be entered if the donor is American and wants a receipt that would enable them to deduct it; 3) below $250, receipts aren't necessary for tax deduction so donor information isn't needed at all.

[1] More precisely, they're required to report all donors who gave more than $5,000 cumulatively in a given year (or possibly a higher threshold for very large organizations), which requires keeping a running sum of per-donor totals. Except that as a concession to ease of recordkeeping, individual donations below $1,000 do not need to be included in the total, and therefore I believe (?) could be accepted anonymously. Or so I read the IRS's guidelines, but IANACPA; see Form 990 Schedule B or your local accountant specializing in nonprofit law for details.

When I used to donate previously, my DKK were converted to USD. But I still use DKK every day.

It matters. Automatically exchanging large amounts of BTC for USD creates a downward pressure on the BTC exchange rate.

Organizations which truly support Bitcoin would choose to keep it.