Question: If Bitcoin is going down, do I only lose if I sell my bitcoin?
So, if I hold them I won’t lose?

Answer: 
Hardware wallets such as the Trezor ones you mentioned are only a secure method of storing and using your private keys. When you want to send BTC, that transaction will be signed mathematically completely within the confines of the key itself.

One *could* design a hardware wallet with a full user interface, negating the need for a separate computer all together, but that would be far more expensive; in the realm of the price of a modern mobile phone. Keeping the code inside a Trezor to the utter minimum reduces the potential for security weaknesses and improves reliability, too.
Note that hardware wallets do you store any bitcoins. Without going too technical, hardware wallets securely store the private keys (derived from the special 12/18/24 words). All your bitcoins are stored in the actual blockchain. You use your private keys to unlock your funds in the blockchain. Think of your wallet as the key and the blockchain as the vault. The hardware wallet does the necessary cryptographic calculations to query your funds or send bitcoin to another address. It cannot do it offline, it must require internet connection to communicate with the blockchain.

> 
> Actually, I vaguely had the idea that I would be able to buy one and
> then physically take it with me somewhere, without any computer
> around, and sort of "beam" any amount of my coins into another
> person's Trezor or even other hardware wallet...
> 
> 
> 

That is technically possible but that means the wallets will require some sort of NFC and also connection to the internet (4G/5G/WiFi). The wallet will also require better CPU and higher RAM. This will significantly increase the price of the hardware wallet and also make it bulkier.

And finally, no, you do not necessarily need to use the Trezor website. You can host the web interface right on your laptop but might be of an overkill for any average user. See <https://wiki.trezor.io/User_manual:Running_a_local_instance_of_Trezor_Wallet_backend_(Blockbook)>