Is the father's intention for the 'aqiqah sufficient when purchasing and slaughtering the sheep in the slaughterhouse, or is it a condition to inform the slaughterer of the 'aqiqah intention?

Your intention for the 'aqiqah is sufficient as long as you have designated the sheep for it. It is not a condition to inform the slaughterer of the 'aqiqah intention, as his intention is not needed. What matters is the intention of the owner of the 'aqiqah. The ruling of the 'aqiqah is like the ruling of the udhiyah in all its regulations, including the intention.