Does mining require downloading the transaction history
You can mine them, as long as you're in a pool, where your earnings (awarded as shares of the blocks you help solve) will go to your pool wallet online until you pull them out into your local wallet.

Generally speaking you want your coins in your local wallet so that if a pool goes down or is compromised in some way, you don't lose your hard-earned coin.

So start downloading the blockchain with your wallet, but while you do so you may also begin mining.