Is Kingdom of Heaven/Kingdom of God the Church?

The New Testament has quite a few parables about the Kingdom of Heaven and Kingdom of God. Is the phrase Kingdom of Heaven the same as the church? My Evangelical Protestant training led me to think that they are not the same, but they are related.

More Dispensational Falsehoods

First, let's look at one Dispensational falsehood. 

Dispensational theology sees a distinction between the Kingdom of Heaven and the Kingdom of God.

As we saw in the introductory article in this series, classical dispensationalism believes that the kingdom of God refers to God’s rule throughout time. The kingdom of heaven, on the other hand, refers to the political kingdom to be set up by the Messiah after his return. Time Periods in the Bible: Questioning Classical Dispensationalism – Kingdom of God and Kingdom of Heaven

Another similar statement:

Knowing the doctrinal difference between the terms "Kingdom of Heaven" and "Kingdom of God" is the key to understanding the complete timeline of Biblical history past, present, and future, the proper place of the Church and the prophetic future of Israel.  Kingdom of Heaven and Kingdom of God The Doctrinal Differences

The first thing to note about this error is that the phrase "Kingdom of Heaven" is only used in the Gospel of Saint Matthew. On its face this creates a serious problem for those who see the two as different. There's just no reason from the text to believe that Matthew was only talking about a future Kingdom and the other Gospels were talking about the Kingdom throughout time.

Parallel passages between the Gospel can be used to easily dispel this notion of a difference. For instance:

Matthew 3:2 And saying, Repent ye: for the kingdom of heaven is at hand.

and 

Mark 1:15 And saying, The time is fulfilled, and the kingdom of God is at hand: repent ye, and believe the gospel.

Many other such passages can be cited:

Matthew 5:3 Blessed are the poor in spirit: for theirs is the kingdom of heaven.

and

Luke 6:20 And he lifted up his eyes on his disciples, and said, Blessed be ye poor: for yours is the kingdom of God.

The Kingdom of Heaven/God Includes Old and New Testament Believers

The Kingdom includes both Old and New Testament saints. They are all present in the same Kingdom.

Matthew 8:11 And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven.

The Kingdom is past, present, and future.  This is demonstrated when Jesus speaks of the Kingdom being present.

Matthrew 10:7 And as ye go, preach, saying, The kingdom of heaven is at hand.

Luke 11:20 But if I with the finger of God cast out devils, no doubt the kingdom of God is come upon you.

The Kingdom of Heaven/God is the Church

In the Gospels, Jesus only spoke of the church (by name) in two passages. The first passage establishes the identification of the church as the Kingdom of Heaven.

Matthew 16:18-19 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

The other passage is:

Matthew 18:17 And if he shall neglect to hear them, tell it unto the church: but if he neglect to hear the church, let him be unto thee as an heathen man and a publican.

This passage can be understood in the ordinary sense of the word translated "church" here as "assembly". From Strong's:

ἐκκλησία ekklēsía, ek-klay-see'-ah; from a compound of G1537 and a derivative of G2564; a calling out, i.e. (concretely) a popular meeting, especially a religious congregation (Jewish synagogue, or Christian community of members on earth or saints in heaven or both):—assembly, church.

So we are left with one of two conclusions. Jesus almost never spoke of the church (in just the two passages mentioned above) or Jesus spoke all the time about the church as the Kingdom of Heaven/Kingdom of God passages in the New Testament.

The church was not a surprise but was the plan of God to have a people called out for Himself all the way along. It was present in the Old Testament.

Acts 7:38 This is he, that was in the church in the wilderness with the angel which spake to him in the mount Sina, and with our fathers: who received the lively oracles to give unto us:

This plan is further detailed by the Apostle Paul:

Ephesians 3:10 To the intent that now unto the principalities and powers in heavenly places might be known by the church the manifold wisdom of God,