I have stock that's depreciated in value since I bought it. Just to make up some numbers, I paid $8000 for it, and it's worth $1000 now. If I donate the stock to charity, can I take the charitable deduction (if I itemize) of $1000 and the capital loss of $7000?
No. You should only donate appreciated stock.
If you own a stock at a loss, you can only deduct the FMV (fair market value) when you donate. Instead, you should sell it, take the loss on your taxes, and donate the cash.
No, it doesn't work like this. Your charitable contribution is limited to the FMV.
In your scenario your charitable contribution is limited by the FMV, i.e.: you can only deduct the worth of the stocks. It would be to your advantage to sell the stocks and donate cash.
Had your stock appreciated, you may be required to either deduct the appreciation amount from the donation deduction or pay capital gains tax (increasing your basis to the FMV), depending on the nature of your donation. In many cases - you may be able to deduct the whole value of the appreciated stock without paying capital gains. Read the link below for more details and exceptions. In this scenario, it is probably more beneficial to donate the stock (even if required to pay the capital gains tax), instead of selling and donating cash (which will always trigger the capital gains tax).
See more here.