Is there a way to import a private key that I have been using in bitcoin-qt to my coinbase wallet?
Coinbase no longer supports any sort of wallet import (wonder why?). To quote Coinbase (June 2016 on https://support.coinbase.com/customer/en/portal/articles/2285419-how-do-i-import-an-existing-wallet-)
Coinbase wallets currently do not support a wallet import feature. If you have paper wallets, private keys, wallet.dat files, or brain wallets, they can be imported into another wallet (Bitcoin QT, Electrum, Multibit, Armory, Blockchain.info, etc) before being sent to your Coinbase wallet. After you have imported your wallet, you can keep your bitcoin there, or send it to one of your Coinbase wallet addresses.
At Import Paper Wallet, you can import a private key. Though it's made for their paper wallet exports, it will work with any valid private key. Using it will transfer the money from the private key into Coinbase, not just allow you to send from it using Coinbase (like adding a private key to Bitcoin-QT would).
I used BitPay's Android app to sweep a paper wallet. You might want to adjust the wallet's fee settings first.
Note that while the BTC got swept, (minus a tx fee) the Bitcoin Cash did not get swept, even though the BitPay app supports Bitcoin Cash and the paper wallet was pre-fork, since the sweep was a new, post-fork tx.
I was able to sweep the Bitcoin Cash from the paper wallet into a Electron-Cash for Mac by manually typing the private key.