Is the church the bride of Christ? Is there a such thing as a Baptist Bride. Is there a universal church that exist now? 

I believe that the institution of the church began during the ministry of Jesus Christ but the organism that includes all believers began about, probably on, Pentecost. I find that this stand is pretty much unsatisfying to everyone. But it satisfies me.

I believe that the church God does His work through today is the local church. That is the emphasis of the New Testament. I believe that each local church is a body of Christ. Paul told the Corinthians, "Now ye are the body of Christ, and members in particular" (1 Corinthians 12:27Now ye are the body of Christ, and members in particular.
See All...). He is speaking to the church of Corinth (1 Corinthians 1:2Unto the church of God which is at Corinth, to them that are sanctified in Christ Jesus, called to be saints, with all that in every place call upon the name of Jesus Christ our Lord, both theirs and ours:
See All...) and he tells them that they are collectively the body of Christ - "ye are."

I also believe that there is a body of Christ that includes all believers in the age of grace. This body is a spiritual organism which as the prospective heavenly church has never met for its first service, but I do see it as an existing body today. However, it is more of a spiritual blessing than an operational unit. Its members are spread over all the earth and in heaven. It is incomplete and finds its final form in the future. This definition of the body of Christ is seen in 1 Corinthians 12:13For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit.
See All... - "For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit. " Here, Paul includes himself in the "we" passage. So he is included in this body and he was not a member of the church in Corinth at this time.

I do not fit in any known group on this one. Paul said that "we" are all baptized into one body. I believe him. Paul told the Corinthians that "ye" are the body of Christ. I believe him. Just as our water baptism is a visible, outward representation of the spiritual baptism that occurred at the point of salvation, so the local church is a visible, outward representation of the spiritual body that will one day assemble in heaven.

As to the bride of Christ, the only spiritual bride that is called such in the Bible is the New Jerusalem (Revelation 21:2And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband.
See All..., 9-10). This, of course, fits almost no one's doctrine. However, the church is compared to a bride in Ephesians 5 and such a comparison is implied in other passages (such as 2 Corinthians 11:2For I am jealous over you with godly jealousy: for I have espoused you to one husband, that I may present you as a chaste virgin to Christ.
See All...). But notice, in Ephesians 5 the church seems to include all believers (since it is the church that Christ gave Himself for). Yet, in 2 Corinthians 11:2For I am jealous over you with godly jealousy: for I have espoused you to one husband, that I may present you as a chaste virgin to Christ.
See All... Paul has espoused the local church of Corinth to one husband. As with the body of Christ, the image of bride seems to fit both the local church of this age and the spiritual body of Christ that includes all believers and will one day be assembled in heaven.