In these sentences, "hope" is a 'bridge verb'; you can use "that" afterwards or omit it. Other 'bridge verbs' include "say", "think", and "believe". For example: "I believe we can save the earth." "I believe that we can save the earth." "I think we can save the earth." "I think that we can save the earth." It depends on the context of your writing, and your exact meaning, but in English we often use the Future Simple tense to talk about unlikely or uncertain events in the future (events with a low possibility of becoming real). Therefore, in my opinion it would be more natural to say: "I hope we will be able to save the earth / I hope that we will be able to save the earth." "I hope people won't fight anymore / I hope that people won't fight anymore."

Yes, they both have exactly the same meaning.