- Existing user which already signed up with social network - should we require an addition of password to the account?
No, that's not in your or the users interest. However, you may add an options for users to create a site specific account. Doing so allows for switching between login method. Sometimes users like to switch account they use to access the service, especially on a paid service.
A user may sign up using Google or Facebook account, only to realize it interferes with an already signed in user on a different device (wife's laptop or daughters iPad). The option to switch between accounts would then help the users to manage their sign in experience.
- New user that is not logged in and is signing up for a trial for the paid version - is it ok to let them signup through social network?
Yes! It's actually a preferred method. Users already manage too many site centric username/password logins. Using a third party identity provider makes it easier for users of the service, and avoid your service to send too many "forgot password" links via e-mail.