This video is provided by @DestructionBomb, and with the description he also provides a great Q&A:
1) What are the attack inputs when you jump over the environments?
The command moves are:
There are 5 moves in total and they are not listed within your move commands.
*While landing* means to hit the attack command when you are about to touch the ground after vaulting in the air, or as you are descending.
2) Will this work after jumping over the sandbags in Hot Zone? and how many stages have this?
Yes. What you see here will work the same way in Hot Zone, just without the open cliff behind that makes them fall off and there is only two stages currently for that environment. (Hot Zone and The Ends of the Earth).
3) I see attacks that looks like it has follow ups after it, do they have follow ups after these certain X/Y moves?
Yes, provided the actual move on the actual command list has string attacks after it. Eliot for example, has his 2H+K converted to -while landing- K here, but the string follow up also returns in this case so the follow up attack would be -while landing- KP. Phase is another example with K and the moment you splat them into the environment. I provided an example on her portion.
If there's anything missing, let me know right away when possible within the comments.
4) What's the purpose of this? what's the point when only two stages has this?
The purpose of this is to provide knowledge on one of the important elements to DOA. The environment. What you see here can be the climax to doing unpredictable damage or to win a close round/match. It's not very often that these environments will trigger, but it's useful to know this as well as showing overlooked/underrated mechanic or attacks that people may not be aware of when placed in this situation.
Note: There is also AI-only specific throws after the iceberg/sandbags jump that the CPU has access to. Unfortunately there is no way to perform them currently as a player.
You can ask Destruction Bomb anything, and he provides a lot of great content via his YouTube covering DOA strategies in a humorous and entertaining way.