It usually works just like with stocks, you found your account and then you can place an order with the fiat on your account and when the trade come through you will have whatever you bought in your account which you can then send to your own wallet.
I have not seen an exchange that lets you buy bitcoin before you give them money but if you know it I would really like to buy a couple 1000 bitcoins, I promise I will send the money after XD.
When you first turn on the trezor (i think ledger does it similarly) you will be prompted to write down a recovery “phrase” which will allow you to recover the wallet to another trezor or to an online wallet (I would only recommend this for recovery). You can always recover another trezor with your key to have to “copies” of the same keys, you can even have different pins.
The trezor itself does actually not store the private key, it only has the “seed” and a repeatable way to generate a private key using this seed. If you look at it that way all you need for another currency is code to generate a private key from the seed in a repeatable fashion (and the whole singing stuff usw).
The tech is actually quite cool, I did not get very far into it but one cool feature about it is that you can have a password in addition to the pin which will be used alongside the seed, this allows you to have “infinite” wallets for the same coin on the same device as behind every possible password there is a wallet. This also gives you plausible deniability as there is no way to know how many wallets are on a device, you are however also screwed if you forget the password.