What is the ruling on prostrating on a thick prayer rug that prevents the fingertips from touching the ground, and should the prayer be repeated if focus (khushoo') is not achieved due to the attempt to press the fingers onto a spongy rug?

There is no harm in placing the toes on the prayer rug and pressing them into it during prostration. It is not obligatory for the toes to directly touch the ground. Your prayer is valid, and you are not required to repeat it. It is sufficient to press your toes into the ground without excessive pressure.