Yes, you can easily move (ERC20) tokens to and from a Ledger Nano S using Myetherwallet.

The Ledger holds your private keys and they stay in this device. This means that even if your computer is hacked and you connect the Ledger, a hacker cannot steal your private keys. Note: It is very important to never enter the 24 word backup seed into a computer (unless you need to recover). And also: do not lose this seed! Once someone has this seed, they have access your coins.

If you know your eth address you can see the balance without your device. This is because the balance is stored in the blockchain.
Receiving coins can be done if someone uses the address you give them and you do not need the Ledger connected for this.
If you want to send coins you will need to connect the Ledger to confirm the transaction you create on your computer with a button on the device.
The Chrome app only shows one eth-address but you have many more. If you use https://www.myetherwallet.com/ and connect your Ledger you can interact with different addresses. If you want to use myetherwallet you need to enable 'Browser support' in the settings of the eth wallet.
The Ledger supports eth tokens like REP, GNT and many more (at least all ERC20 tokens). To send tokens to the device you send them to one of the eth-addresses. To send token from your device you need to enable 'Contract support' in your Ledger settings and use myetherwallet. You select the eth-address you sent the tokens to and choose if you want to send tokens. NOTE: you will need to have (a small amount of) eth in the address to send tokens! Check with small amounts first! When sending choose the correct token and not eth.
When you first set up the Ledger after you've recorded your seed, reset the Ledger by entering an incorrect PIN 3 times and recover using the seed before sending any coins to ensure you recorded the seed correctly.
When sending coins for the first time, send a very small amount to ensure you have the correct address.

Addition about backing up your seed:

The way you store the 24 word seed is up to you. You can write it on a piece of paper and hide it very well or split it in two 12 word parts and store them in different locations. You can also use steel like for example cryptosteel and put it in a vault somewhere. Or put it in your wall or bury it :)