What is the ruling on a Muslim performing acts of worship with the intention of having prayers answered and worldly affairs facilitated? And can sins be an impediment to the answering of prayers?

One must be content with Allah's decree and destiny, for what Allah has chosen is the best, and there may be great good in the loss of something.

One must not surrender to the whispers of Satan, but rather thank Allah for His blessings, be content with His decree, and busy oneself with obedience to Allah and a comprehensive devotional program, remembering the Prophet's (peace be upon him) saying: "Wondrous is the affair of the believer, for all his affairs are good."

One must lighten the burden on oneself and remember that many unmarried women are happy, and that seeing married women should not evoke feelings of deprivation or injustice, for married life may entail difficulties and tragedies not visible to the naked eye.

You mentioned the story of a woman who reached forty and did not marry, and how she found happiness and contentment by drawing closer to Allah, reciting and interpreting the Quran, and performing good deeds.

The purpose of life is to worship Allah. If marriage is not available, then there are many general ways of worship, foremost among them being calling to Allah, spreading guidance, and knowledge.