Each time I start Windows, I forget to start True Key until I need it at some point. Is there a way of getting the login screen to appear during Windows startup? Also, when I do double click the True Key icon - on my laptop is already shows my email address, but on my desktop I have to click the 'sign in' link.
What you could since you are using Win10, is to utlize Windows Hello to log in (using either a pin, fingerprint or picture), just type Windows Hello on the bottom left of your desktop and in the search bar enter Windows Hello, you should be offered a few sign-in options.
Once this is done, you can go into True Key settings and linked this newly Windows Hello profile to your True Key account.
Once this has been added, you should not have any issues using Windows Hello to log in with the TK icon at the Windows login page, and you won't be prompted to enter your TK credentials anymore unless your sign-out from your system, or due to the True Key sign-out time.
Let me add one more thing, since I am not sure the browser you are using, if it is Edge, it might also empyting the cache (which includes your login credentials to True Key) as soon as you exit the browser, this might explain why the different behaviour between your PC and your laptop, just because you are using different browsers or True Key extension first (only on the browser) vs. True Key standalone (on the desktop).
I hope these clears some of the questions you had.