Practically, all D-mail, time leap and time machine are sent to different worldline as the wordline has already changed at the moment they arrived. But in either case, your question have a point.
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how could Future Okabe have sent a video message across timelines?
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It is mentioned only in the novelized version "Steins;Gate Enkan Rensa no Ouroboros". The logic is complex, so to write only about conclusions, future Okabe invented some technology called Worldline Compression based on Worldline Duplication theory, and by using it, he spent more than 100 years to send countless segmented video D-mails at countless different wordlines. In most worldlines the mail was received as incomplete video file and became noise. Finally in the worldline of episode 23, it was recieved as a complete video mail.
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Did the timeline change after Okabe saves Kurisu in the TRUE END (I'm assuming it did since Suzuha disappeared on the time machine), and if so, why was Okabe hospitalized (only his mind should have traveled, so his wounds shouldn't exist anymore)?[B]
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Yes, the worldline changed to Steins Gate worldline where no future was pre-determined, which means time machine would never be invented. How the past of the Steins Gate worldline was rebuild hasn't been told yet. All we know is Kurisu was saved by Okabe for some reason and then Okabe was hospitalized for some reason. You can call it plot hole, or "it is left up to viewers" or "will be told in my fanfiction!" or anything.