I scrolled down the page and noticed that you have an available app for mobile users. I decided to download it so that I'd be able to check my answers and also to answer some questions through my mobile. However, as soon as I downloaded the app, I tried to log in my account, only to realize that there is no function of logging in by Facebook, so how am I supposed to log in my account since there is no such feature? Is there any other way? My account is only accessed through Facebook — is there any way to create a password to the account so that I can log into my account through that pass?
Yes, you can follow the first procedure in the Help Center when on the website to add an OpenID login. In particular, consider using Stack Exchange's own OpenID provider.
I added a new login method through Google, so I downloaded the app into my mobile, When I want to log into my account through the mobile, should I use the Google account which I associated my account with?– DavydDec 31, 2016 at 12:24
Yeah, that should do the job. Dec 31, 2016 at 19:37
There is a discussion on the overall StackExchange Meta that explains that you have to have the official Facebook app installed:
Login with Facebook using Stack Exchange Android App
I don't use Facebook, so I can't test to see if anything has changed since 2014 when the question was was asked but I think it should work.