If you have the Ledger Nano S or a Nano S Plus, then your device needs to be connected and plugged into your computer to be able to send transactions. On the other hand, if you have the Ledger Nano X that has Bluetooth connection and a 100 mAh battery(several hours of active usage, a few months when idle), you can send transactions simply by connecting your device to your mobile phone via Bluetooth.
No, your device doesn’t need to be connected and plugged in to be able to receive payments. As long as you have a copy of your wallet address, people can send funds to your wallet as much as they want. The same applies to any other wallet, may it be a Ledger wallet, or a software wallet.