On smaller screens I think some UI elements will have to be out or subset in a menu, 'cos there's a lot on the screen now. A good start, in my opinion, is to match Apple design guidelines when they say to design for the 80% majority, which in this case means that commands used by 20% of the people could be put out. Viewing the main screen is always more important than having a non-standard option.
Buttons like the "view all levels" and "help" could be cut out.
This is kinda what I was thinking as you saw before. I'd do one thing different though: I'd replace the "abort mission" button with the options button. That's what the ? button does (not "help"), it opens up the game pause menu. Abort Mission could be put in there. However, every device (even iOS) has a 'Back' button that could call that instead of a soft button, so that could just become free space for something else (confirm shoot/move maybe?)
I suppose for the sake of mitigating fat-fingering, the End Turn button could have a confirmation dialog too.
For that above, I don't know if this was covered already, but the ideal case is to have a simpler version for smaller screens aside with a complete version for bigger screens. All good system/apps with multiple versions do so: the smaller the screen, the simpler the interface (simple in this case means 'with less features').
This would be ideal, but I don't think OpenXcom is currently designed to be able to dynamically change this kind of thing, since I believe most UI is hardcoded right now. But I suppose there could just be two versions of the game for that (a Tablet and Phone version)
The soldier stats and rank could be simplified or cut to open space. In the stats case, for the sake of briefness you could leave the numbers and cut the bars or, if possible, change the bars into something smaller.
This is one thing I somewhat disagree with. Shortened, maybe, but I don't think the bars should be totally gone. It's pretty hard to estimate just how much of your soldier's health is left when it's just a number with no visual indication. Though for TUs it would be fine.
Maybe it could display the total number next to it, so HP could show as "18 / 25" or so instead of just 18, if bars were to be totally cut.
A clever idea is to eliminate all back and forth buttons (up and down too) that can be looped (level viewing, prev/next xcom operative).
Nice one, I hadn't actually considered this for level viewing. But what about move up/down? In your design, that's only one button. That can't wrap, especially because each move takes TUs. Does your button bring up a dialog that asks which way to move?
Also, Next/Prev unit wraps too, I only added "Prev Unit" because I wasn't sure what else to do with that space. It wasn't in the original game.
I don't think a toggle button with all four options (like the TU saving mode presented) is a good idea. People will try to press only the part they want, making it uneffective as a big button. Two ways to make it work: show only the current mode or don't use green background, but a html-radio-like above the options.
( I don't get why the crouch option have to be apart, but I must have bypassed it here... )
Good point, having it as a single cycling button would work best. Mostly because html-radio-like callouts aren't implemented in Xcom in any sort of way. But it could use the OS to do that, I suppose.
Also, the crouch reserve button was separate because it works separately to the other ones (you could opt to reserve TUs for both shooting and then crouching after shooting, for example) It does take up space so it could be moved to long-pressing the regular crouch button (though this isn't quite as intuitive).
I honestly never used it, so I wouldn't mind not having it (it wasn't there until TFTD anyhow)
Last, but not least, the soldier stats screen and the soldier ground screen could be called by pressing the soldier name/stats and both screens could have a button to call the other screen (stats/ground).
The inventory screen can actually already be called from clicking the rank icon in the inventory screen! So yes, that works.
Otherwise, your proposed layout is great. I really didn't like having to put buttons at the top of the screen...
EDIT: So about the 'confirm shoot/move' button.
I was thinking your Move Unit Up/Down button could be replaced with a 3-way control mode toggle, it'd have:
Move Mode: Move unit around.
Look Mode: Turn unit to face things.
Move Up/Down mode: Swipe unit up/down to move it up/down a level
That way the problems with having to tap the cursor to confirm a move/shoot could be avoided, and the button to the right of it could be replaced with a "Confirm" button.