Split the first sentence in two*:
Keep both your feet flat on the floor
Keep both your feet not
pulled back under your chair.
You can see that you've got two modifiers of "feet" -- "flat" and "pulled" back, which is to say a compound structure in the original sentence, with the comma replacing the conjunction.
Keep your both feet flat on the floor which are not pulled back under
doesn't quite work because you've lost the sense of the double command: do one thing and not the other.
In the second sentence
Do you lead a sedentary life, slumped in a chair all day?
the participial phrase is best understood as a nominative absolute, standing free of the rest of the sentence, applying to both subject and predicate. The word "slumped" describes both your bodily arrangement and how you face your life. Your rephrase is fine.
* I've taken the liberty of transposing "your" and "both".