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.
Examples:
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.
With "Why would you do that?" you're questioning the other person's motivations, which implies that a) it was deliberate and b) they might do it again.
With "Why did you do that?", you're asking about the decisions and circumstances around that past event.
Let's say you broke an ornament (which you hate) by accident, and your partner says "Why would you do that?", they're implying you did it on purpose. You might say "I didn't mean to, it was an accident!". You're replying to the implication that it was deliberate.
If they say "Why did you do that?", you might say "I tripped on one of the kids toys and knocked it off the shelf." Here you're just saying what happened.
EDIT: this answer assumes that the questioner is asking about something that has happened. If the thing being discussed is hypothetical then as @EdwinAshworth points out in his comment, it takes on a less accusatory tone: because nothing has actually happened, there's no question of deliberate vs accidental for example: you're simply asking about someone's motivation or opinions.
Best Answer
Can you please give the source?
Your sentence is
"that" here stands for "motive". So rewriting the sentence with he replacement of "that" we find
Here are a few more example sentences which have similar kind of construction -
Is it fine now?