No. We should not pray for our enemies to die. We are to love our enemies. Matthew 5:44-45, "But I say to you, love your enemies, bless those who curse you, do good to those who hate you, and pray for those who spitefully use you and persecute you, that you may be sons of your Father in heaven; for He makes His sun rise on the evil and on the good, and sends rain on the just and on the unjust." But, perhaps you are afraid of what your enemies might do to you.

If you have accepted Jesus as your savior and have consecrated your life to do God's will, then every single thing that you experience is either over-ruled or permitted by God to bless and to develop you. God loves you so much that He even knows how many hairs are in your head. Luke 12:7, "Indeed, the hairs of your head are all counted. Don’t be afraid…" 

We should have faith that should an enemy hurt us, that God has over-ruled it for our good. That probably sounds strange. But Jesus, himself, said, [Mat 5:10-12, "Blessed are those who are persecuted for righteousness' sake, For theirs is the kingdom of heaven. Blessed are you when they revile and persecute you, and say all kinds of evil against you falsely for My sake. Rejoice and be exceedingly glad, for great [is] your reward in heaven, for so they persecuted the prophets who were before you." Matthew 5:10-12.

Are we to pray for Satan’s destruction? We could. It would show faith in God’s promise to destroy Satan. Ezekiel 28:19, “All who knew you among the peoples are astonished at you; You have become a horror, And shall be no more forever.” Satan’s complete and final destruction will not occur until the end of Christ’s 1,000 year reign on earth. (See Revelation 20:10.)

Finally, if the church you are attending does not follow these scriptures, then you need to find a different church.