Do all my sentences sound natural?
"I pray every day that such a thing doesn't happen."
"I pray every day that such a thing will not happen." (Or "will never happen").
"I hope I will never face such a situation."
Going over this in my head, "such a thing" is a usage that seems to have been much more common fifty years ago. If a person was just saying this in conversation today, I think they would phrase it as:
"I pray every day that doesn't happen." or
"I pray every day that this doesn't happen."
or, and I know this one is a little silly, but English IS silly, "I pray every day that that doesn't happen."
When saying this sentence, it is always going to be proceeded by talk of the feared event, so one doesn't need to say "such a thing," it is clear what the person is talking about.
If you really wanted to dig in, it should be noted that putting the stress on "pray" would be common in the Southern United States, but in more urban environments, the stress would be on "every day." Such as:
"Every day I pray that that won't happen."
I would also just add that my examples are conversational English. If you are writing answers for a homework assignment, the way you already have it is probably best.
Those all sound completely natural to me, a native speaker. "Pray" sounds more emphatic than "hope", and does not / will not are interchangeable here.
Everything sounds nice to me.
Just the second one, I think that like this is better:
"I pray every day that such a thing will never happen"
I find the present tense unnatural (and somewhat illogical; you're discussing future events), but it is common usage by native speakers.