Hi, wollern,
I’ve migrated yesterday to Bitwarden from Lastpass and I’m too on Firefox browser on PC (but Chrome on iPhone), so I may have same similar issues/questions. Here is what I know so far:
(1) It is possible to use it in Bitwarden add-on, but they say it’s an experimental feature. To turn this feature on go and click in Firefox on the add-on -> go Settings -> Options and there should be, on the first line, “Enable auto-fill” check-box. If it is not there switch your browser to English (to switch Firefox go to Option - Preferencies, or put about:preferences into URL in Firefox), because I saw this check-box only in English version. After I checked that and went back to my default language the check-box was now available in the default language and the auto-fill feature stayed checked, so you don’t have keep Fifefox in English if you don’t want to.
(2) It was not asking me to save my password either. But after I turned on the autofill function as described in (1) it asks me now. So, maybe these functionalities are connected and mutually conditioned, I don’t know…
(3) To generate new password during new registration on some website I click right mouse in the password field, select Bitwarden from the context menu and click “Generate password and save to clipboard” (or what ever it says in your language). So, the password gets generated and is saved in my clipboard, therefore I then only do CTRL+V in the password field and it is filled in there and when successfully logged in the add-on asks me to save the password to the vault. I use the same settings for all my passwords (the lenght, A-Z, a-z etc.) so I don’t need to adjust the generator in the add-on manually before generating new passwords.
Not sure if this helped you with all your questions, but I hope it might a bit 
Tomek
PS: Don’t forget to also copy your Equivalent Domains list from Lastpass to Bitwarden: Here is how to