Why do Protestants hold "prayer services"?

From a Fundamentalist Pentecostal pov, "Prayer Services" are usually geared towards corporate prayer and petitions of God.

These prayer(s) can be very long (over 30mins each). If any hymns are sung or preaching is preached, it is to prepare the church (congregation) for prayer.

These prayers can be in support of a specific cause; i.e. imprisoned missionaries, warring countries, illness, etc.

This is different from "regular worship services" where the focus is preaching/teaching.

**EDIT: In the quest to find Biblical standing for "Prayer Meetings", it doesn't seem like there any passages of Jesus explicitly being part of a "prayer meeting", since majority of the time he spent that time alone with God and sometimes invite a few to support Him.

Here are the passages that talk about him being alone:

Luke 15:16 (NASB)
But Jesus Himself would often slip away to the wilderness and pray.

Luke 22:41 (NASB)
And He withdrew from them about a stone’s throw, and He knelt down and began to pray

Mark 1:35 (NASB)
In the early morning, while it was still dark, Jesus got up, left the house, and went away to a secluded place, and was praying there.

Matthew 14:23 (NASB)
After He had sent the crowds away, He went up on the mountain by Himself to pray; and when it was evening, He was there alone.

Luke 6:12 (NASB)
It was at this time that He went off to the mountain to pray, and He spent the whole night in prayer to God

Christ Himself advocates that we pray in isolation:

Matthew 6:6 (NASB) But you, when you pray, go into your inner room, close your door and pray to your Father who is in secret, and your Father who sees what is done in secret will reward you.

It's possible that the Apostles congregated for prayer:

Acts 1:12-14 (NASB)
Then they returned to Jerusalem from the mount called Olivet, which is near Jerusalem, a Sabbath day’s journey away. When they had entered the city, they went up to the upper room where they were staying; that is, Peter and John and James and Andrew, Philip and Thomas, Bartholomew and Matthew, James the son of Alphaeus, and Simon the Zealot, and Judas the son of James. These all with one mind were continually devoting themselves to prayer, along with the women, and Mary the mother of Jesus, and with His brothers.

Other followers of Christ praying together for a cause

Acts 12:12 (NASB)
And when he realized this, he went to the house of Mary, the mother of John who was also called Mark, where many were gathered together and were praying.

Summary of the Biblical references
Although I've been unable to find passages that explicitly talk about Jesus praying with His disciples, we do find the it was done by the early church.

This answer was written in answer to the original question, before it was massively rewritten and as the question is now asked does not apply to or answer your question.

Please do not down vote or otherwise denigrate my answer, since it does apply and does answer the original question.

Your question cannot be answered simply. In answering your question we must first know what is prayer? Prayer is much more than just talking to God. There are in fact many different types of prayers, Intercessory prayers, personal prayers and so on.

All Scripture is quoted from the King James translation:

Some prayers need to be personal such as when Jesus said to his disciples;

Matthew 6:6 But thou, when thou prayest, enter into thy closet, and when thou hast shut thy door, pray to thy Father which is in secret; and thy Father which seeth in secret shall reward thee openly.

Personal prayers should be just that, they should be between you and your God. When you are asking God for a particular blessing, such as the perfect mate, you are in fact asking God amend his plans and alter things in order for you to satisfy your personal situation. While it is not wrong for you to make such a petition, it is not something that should be requested in public.

Just as there are some prayers, that should be private there are some prayers which should be shared, intercessory prayer is one in which it is necessary that those being prayed for should be aware of it. Jesus taught us this with his intercessory prayer for the Disciples in:

Joh 17:1 through 26 These words spake Jesus, and lifted up his eyes to heaven, and said, Father, the hour is come; glorify thy Son, that thy Son also may glorify thee: As thou hast given him power over all flesh, that he should give eternal life to as many as thou hast given him. And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent. I have glorified thee on the earth: I have finished the work which thou gavest me to do. And now, O Father, glorify thou me with thine own self with the glory which I had with thee before the world was. I have manifested thy name unto the men which thou gavest me out of the world: thine they were, and thou gavest them me; and they have kept thy word. Now they have known that all things whatsoever thou hast given me are of thee. For I have given unto them the words which thou gavest me; and they have received them, and have known surely that I came out from thee, and they have believed that thou didst send me. I pray for them: I pray not for the world, but for them which thou hast given me; for they are thine. And all mine are thine, and thine are mine; and I am glorified in them. And now I am no more in the world, but these are in the world, and I come to thee. Holy Father, keep through thine own name those whom thou hast given me, that they may be one, as we are. While I was with them in the world, I kept them in thy name: those that thou gavest me I have kept, and none of them is lost, but the son of perdition; that the scripture might be fulfilled. And now come I to thee; and these things I speak in the world, that they might have my joy fulfilled in themselves. I have given them thy word; and the world hath hated them, because they are not of the world, even as I am not of the world. I pray not that thou shouldest take them out of the world, but that thou shouldest keep them from the evil. They are not of the world, even as I am not of the world. Sanctify them through thy truth: thy word is truth. As thou hast sent me into the world, even so have I also sent them into the world. And for their sakes I sanctify myself, that they also might be sanctified through the truth. Neither pray I for these alone, but for them also which shall believe on me through their word; That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me. And the glory which thou gavest me I have given them; that they may be one, even as we are one: I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me. Father, I will that they also, whom thou hast given me, be with me where I am; that they may behold my glory, which thou hast given me: for thou lovedst me before the foundation of the world. O righteous Father, the world hath not known thee: but I have known thee, and these have known that thou hast sent me. And I have declared unto them thy name, and will declare it: that the love wherewith thou hast loved me may be in them, and I in them.

When we take a longer look at this prayer we find that even though it is to the Father, It is really for and directed to the Disciples. Most of what is being said is not for God's benefit, but for the disciples. God was supremely aware of the points Jesus was making.

We need be mindful of this when we pray for others.

Now to answer your question about corporate prayer.

There are situations in which we are directed to pray for General well being.

1st Timothy 2:1 through 3 I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men; For kings, and for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty. For this is good and acceptable in the sight of God our Saviour;

We are given an example of this in type of group prayer in the passages where Peter is loosed from Prison by an Angel:

Act 12:12 through 17 And when he had considered the thing, he came to the house of Mary the mother of John, whose surname was Mark; where many were gathered together praying. And as Peter knocked at the door of the gate, a damsel came to hearken, named Rhoda. And when she knew Peter's voice, she opened not the gate for gladness, but ran in, and told how Peter stood before the gate. And they said unto her, Thou art mad. But she constantly affirmed that it was even so. Then said they, It is his angel. But Peter continued knocking: and when they had opened the door, and saw him, they were astonished. But he, beckoning unto them with the hand to hold their peace, declared unto them how the Lord had brought him out of the prison. And he said, Go shew these things unto James, and to the brethren. And he departed, and went into another place.

The best reason for group prayer is given in:

2nd Chronicles 7:14 If my people, which are called by my name, shall humble themselves, and pray, and seek my face, and turn from their wicked ways; then will I hear from heaven, and will forgive their sin, and will heal their land.

Most of us hope that we will be joined by our fellow Christians, who are God's people and called by his name, in constantly doing this.