" I noticed that sometimes in General Chat, a red number flashes."

-Carol,i have never noticed it," red number"?
Oh Do you mean the number in red circle?for instance,when it shows "1" = someone just wrote 1 thing in chat,"2" = two things just got written in chat ...and so on.
After the number "9" it will show the sign "!" .
But if the chat window is open for you,then these numbers won't appear,cause you can see what's going on in chatroom.

"Also, is the chatroom only for people who want to teach each other Russian/English, or is it also a socialization room?"

- I am new here,but the first time i was here,i asked one person my problem in private chat,and i also thought it was only for asking language questions.but after watching General Chat a little,i saw users talk about everything ( including Language!)
So yes,General Chat is an Amazing place with a lot of Great users ,to talk about Anything,while improving your language skills.

"...but I still want to chat with Russians who speak pretty good english"

(Ed. ⇝English ,'E' )
-we have many Russians who speak English greatly there.come and see!