Your questions are quite reasonable and fair to ask. I myself wrestle with questions about God's intervention in human affairs and even, at times, about God's existence and my ability to percieve him despite my profession of being a Christian.
Well, I believe, the answer to your first question: Does prayer change God or His mind? The answer is YES.
Reasons for my answer, We can be confident of God’s response to our prayer when we submit first to his will.
13Are any among you suffering? They should keep on praying about it. And those who have reason to be thankful should continually sing praises to the Lord.
14Are any among you sick? They should call for the elders of the church and have them pray over them, anointing them with oil in the name of the Lord. 15And their prayer offered in faith will heal the sick, and the Lord will make them well. And anyone who has committed sins will be forgiven.
16Confess your sins to each other and pray for each other so that you may be healed. The earnest prayer of a righteous person has great power and wonderful results. 17Elijah was as human as we are, and yet when he prayed earnestly that no rain would fall, none fell for the next three and a half years! 18Then he prayed for rain, and down it poured. The grass turned green, and the crops began to grow again.
1 John 5:14
14And we can be confident that he will listen to us whenever we ask him for anything in line with his will. 15And if we know he is listening when we make our requests, we can be sure that he will give us what we ask for.
Sometimes, like Paul, we will find that God answers prayer by giving us not what we ask for but something better.
2 Corinthians 12:7-10
7even though I have received wonderful revelations from God. But to keep me from getting puffed up, I was given a thorn in my flesh, a messenger from Satan to torment me and keep me from getting proud.
8Three different times I begged the Lord to take it away. 9Each time he said, “My gracious favor is all you need. My power works best in your weakness.” So now I am glad to boast about my weaknesses, so that the power of Christ may work through me. 10Since I know it is all for Christ’s good, I am quite content with my weaknesses and with insults, hardships, persecutions, and calamities. For when I am weak, then I am strong.
And finally, our prayer must be accompanied by a willingness to obey with our actions.
15Then the LORD said to Moses, “Why are you crying out to me? Tell the people to get moving!
As for your second question, it would not make any real sense to pray to a god who was not listening to you or anyone else. One would or could adopt the attitude that agnostics have, that there is a God but he is unconcerned with human affairs.
As for myself, I have come not to trust my perceptions in the matter too much. Because of many failled expictations in my life,( mostly due to religious leaders teaching me lies ), I have adoppted more of a waiting and introspective attitude concerning my prayers to God, whom I firmily believe exist according to the Bible. I am not too quick to believe that God has or has not answered a particular prayer. My prayers tend to be more generalized but also spacific at times. Health and wellbeing for family, friends, believers worldwide and such. Things like a spacific new car or not having to face a bad situation I usually leave up to God's guidence because I believe He knows best in such matters.
Anyway, I hope I've made some sense.
1 Peter 3:12
12The eyes of the Lord watch over those who do right,
and his ears are open to their prayers.
But the Lord turns his face
against those who do evil.”