I have my coins on Electrum at the moment and am happy, but wanted to get a cold wallet for added security*
Once you set up your new Trezor hardware wallet, you will need to move your coins from your Electrum software to an address on your Trezor. After that you can use Electrum as the software for sending and receiving bitcoin as DireWolfM14 explained.
2. As a newbie, I was surprised that there is no advice warning you if you should stay offline at any point when setting up all the password, seed and PIN information.
You don't need to and probably you can't because you need an internet connection to fire up the Trezor software. One of the benefits of hardware wallets is that you can use them even on computers that are infected with malware. Your private keys will remain safely stored on the device.
3. I know this seems weird, but there is no information on how to log out of Suite! Do you just click the x at the top right-hand of the screen?
I don't use Trezor, but I don't think it should be any different than when using Ledger Live or Electrum. You don't log out from those either, you just close the app. Ledger Live does have an automatic log out feature that logs you out automatically if the app is inactive for 10 minutes or more. Check the settings of your Trezor Suite, maybe they have something similar.
If your hardware wallet isn't connected and unlocked, you don't need to be concerned with having the Trezor software open. You have to sign all transactions manually by pressing buttons on the device. Without that, nothing works. If you share your computer with other people, they could see your portfolio if the app remains active. That's one concern.