In SSB4, how do you control whether your character lands on their back or their stomach after falling? It seems that characters have different get-up-attacks (press A while laying on ground) based on whether they are laying on their stomach or their back after a fall so it would be nice to be able to control which one occurs.
It depends on the angle of the character at the exact moment he's touching the ground: you can check it in Training mode in ¼ speed.
Since mosy actions you can do while tumbling will stop the tumbling animation and will prevent you to lay on the ground, the only options you have to control it are fast falling, reducing the knockback by crouch-cancelling, or via directional influence.
And you're right that their get-up attacks are different whether they were laying on their stomach or their back: their hurtboxes are different, and so are their invincibility frames.
Magus420 did a great work compiling various data in Super Smash Bros. Melee, and even if all the frame data are not the same in Smash 4, it can help you to understand all the possibilities you have while laying on the ground.