You wouldn't fill out a 1099, your employer would or possibly whomever manages the stock account.

The 1099-B imported from E-Trade says I had a transaction with sell price ~$4,500.

Yes. You sold ~$4500 of stock to pay income taxes. Both the cost basis and the sale price would probably be ~$4500, so no capital gain. This is because you received and sold the stock at the same time. If they waited a little, you could have had a small gain or loss.

The remainder of the stock has a cost basis of ~$5500.

There are at least two transactions here.

1. Receive ~$10,000 worth of stock. Cost basis of ~$10,000.
2. Sell ~$4500 worth of stock. Cost basis of ~$4500.

In the future you may sell the remaining stock. It has a cost basis of ~$5500. Sale price of course unknown until then. You may break that into different pieces. So you might sell $500 of cost basis for $1000 with a ~$500 capital gain. Then later sell the remainder for $15,000 for a capital gain of ~$10,000.

You wouldn't fill out a 1099, your employer would or possibly whoever manages the stock account.

The 1099-B imported from E-Trade says I had a transaction with sell price ~$4,500.

Yes. You sold ~$4500 of stock to pay income taxes. Both the cost basis and the sale price would probably be ~$4500, so no capital gain. This is because you received and sold the stock at the same time. If they waited a little, you could have had a small gain or loss.

The remainder of the stock has a cost basis of ~$5500.

There are at least two transactions here.

1. Receive ~$10,000 worth of stock. Cost basis of ~$10,000.
2. Sell ~$4500 worth of stock. Cost basis of ~$4500.

In the future you may sell the remaining stock. It has a cost basis of ~$5500. Sale price of course unknown until then. You may break that into different pieces. So you might sell $500 of cost basis for $1000 with a ~$500 capital gain. Then later sell the remainder for $15,000 for a capital gain of ~$10,000.