It will scan the blockchain to load your wallet's transaction history, but won't download the entire blockchain like a full Bitcoin client does. There is also an option to set the date of the earliest transactions in your wallet if you don't want to scan the entire blockchain.
That's not how it works. If you are not using your own node, then it connects to a third party server and queries it for transaction history of your addresses and wallets. The third party returns the necessary data, but you therefore have zero privacy from that third party.
If you want to use it in a privacy preserving way with your own node or Electrum server, then you will absolutely need your node to download and sync the full blockchain first.
You are absolutely right, node must be in fully sync state but I think @logfiles meant that Sparrow has an option to query it for transactions of its interest beginning from certain date rather to initiate the full scan of LevelDB kept by node. In this respect he is correct.