In 1, my hope is occurring now, regarding an event in the present (that you are okay.) It could also refer to an event in the past or future.
You're telling me a story about a dangerous activity your brother engaged in in the past, and before you finish I interrupt with:
I hope that he was okay.
You called to tell me that your brother has crashed his car. I reply:
I hope that he is okay.
You tell me that your brother is going to have to have heart surgery. I reply:
I hope that he will be okay.
In all three versions the action of me hoping occurs at the time that the sentence was spoken, but the event that my hope is focused on need not happen at the same time.
In 2, "are hoping" is used to say that the action, hoping, is continuous. That is, the detectives hoped that witnesses will come forward and they continue to hope that witnesses will come forward. Here, because the action is continuous, it will usually refer to the future, though it could refer to a present (possibly continuous) action as well, or a past event whose outcome was uncertain.
She is hoping that her brother recovers quickly after surgery.
She is hoping that her brother recovers/is recovering quickly after surgery.
She is hoping that her brother recovered quickly after surgery.
In all three the action of hoping is continuous, that is, she hopes now and will continue to hope in the near future.