I'm coming off as possibly annoying for questioning my friend's wellbeing
I think that it shows that you care since you are asking about it!
OK...but do you think I'm annoying?
No I do not think that is annoying. It is nice that you are that concerned about the well being of your friends.
Do you think my friend is okay because of how she acts?
I am not sure, I think that is something that you should talk to your friend out. As far as I am concerned I think that it is okay to inquire about how your friend is doing.