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She's got these stock certificates. There's no way for her to buy/sell anything unless she opens and account with a broker, correct?

Not necessarily. Many companies these days - maybe even most - have stock purchase and dividend reinvestment plans. If she's still interested in reinvesting dividends, she could enroll in the company plan and send them her certificates (for that company, obviously). If she's in such a plan, she can liquidate part/all of her holdings at any time, at little or no cost, through the plan administrator.

So, I'm thinking I should advise her to open an account with a discount brokerage firm and transfer the certificates to them.

But yes, more generally she would be wise to open an account with a discount brokerage. Several of them offer very small commissions and often no fees at all.

It looks like her old broker was routinely selling stock in order to generate the cash for her quarterly fees. Also, there are some mysterious random sales, as well as liquidation of fractional shares on the last statement she received from her old broker. So, I think there are some gains that she needs to report.

I don't know about the mysterious random sales, but the liquidation of fractional shares shown on the last statement is perfectly normal. They only issue certificates for whole numbers of shares, so when you ask for a certificate (as your girlfried did) they sell the fractional part. And yes, all these sales need to be reported, and taxes paid on gains, if any.

When I'm figuring her cost basis for the shares that were sold, can I pick which purchase-times/costs I want to use, or do I need to combine them all for an average cost basis?

No and no. Absent any arrangement with the broker to sell specific lots of shares, you must use FIFO (first in, first out). That is, the earliest shares purchased were the ones sold. And average cost basis works only with mutual funds. You can't do that with stocks.

Due to the dividend-reinvestment, she has some short-term and some long-term holdings.

Again, each sale is a separate transaction. The shares sold were the oldest ones she had at the time of the sale. Whether it's a short or long term transaction depends on the difference between the purchase and sale dates. Remember, you can aggregate purchases and use a date of "various" - but you do have to keep short and long term sales separate. A single sale can have both a short and long term component, in which case it is broken down as two separate sales.

Also, am I correct in thinking that you don't need to pay capital gains tax on dividends that you re-invest?

Dividends, whether reinvested or not, are taxed as ordinary income. When you eventually sell shares acquired through dividend reinvestment, you will have a capital gain/loss, and if the former, you'll pay tax on it. But it's not a tax on the dividend - you already paid that! It's a tax on the gain realized subsequently.

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