Videos on youtube don't really tell the proper story, and reddit seemed to give contradictory answers.
Here's what I surmised:
Kled+Skaarl have separate health bars:
Damage: Skaarl takes damage first (making Kled immune). Any debuffs on Skaarl death are cleared and Kled gets a clean slate
Heals: Skaarl gets all heals first, then when he's full, Kled heals after.
For the purpose of executes, or skills that do bonus to missing health, Kled + Skaarl together form their "total health amount". So if Kled is 800/800, and Skaarl is 1/1600, together, they are at 800/2400 (or about 33%). So ults like Jhin, Garen, Veigar will deal bonus damage according to this 33%.
For full executes like Urgot, Urgot's ult will proc if total health is under threshold -- if Skaarl is alive, Kled lives and gets dismounted, otherwise, Kled goes through the blender. (There are videos on that).
For Pyke (no one made videos of this, perhaps everyone stopped playing Kled), same idea as above... but will Pyke get a reset/gold if he 'executes' Skaarl? I'm guessing not.