Does "faith" enable "love"? Romans 4:16 says:  Therefore [it is] of faith, that [it might be] by grace…

In this verse it appears that “faith” enables “grace”.  The case could be made that the initial breach with God was one of  ”faith”, mankind placed their faith in the "word" of the serpent rather than in the Word of God.
Also Acts 14:27 speaks of “the door of faith”.

My question: Is there any early church teachings which link “the door of faith”, in which mankind exited a relationship with God to the “door of faith” wherein mankind enters into a relationship with God?  And does "faith" enable a loving relationship?
From what I know of Augustine, he would say that, rather, God's love is the enabler of our faith. That is, God's love for the sinner is the ultimate cause of creating new life in the believer, which enables that believer to have faith. 

See Augustine's On Grace and Free Will, Chapter 17. And if you look at Chapter 12 and 13 of the same work, you will see that he says that out of love God shows mercy and grace, which links God's love for us to faith. 

In Augustine's Enchiridion, Chapter 8, he says


  Now what shall I say of love? Without it, faith profits nothing; and
  in its absence, hope cannot exist. The Apostle James says: "The devils
  also believe, and tremble." — that is, they, having neither hope nor
  love, but believing that what we love and hope for is about to come,
  are in terror. And so the Apostle Paul approves and commends the
  "faith that works by love;" and this certainly cannot exist without
  hope. Wherefore there is no love without hope, no hope without love,
  and neither love nor hope without faith.


For Augustine, then, faith, hope, and love are a package deal. The general pattern here, overall, is that God shows us love, which gives us faith, hope, and love towards others, including God. 

As to the relationship between the "two doors", as you've called them, again, I think Augustine would say that the exit door is a trap-door - a one-way door. Closing that door means you do not have the ability to open it again. Instead, God must open it for you. See On Grace and Free Will, in Chapter 7, where he says


  ...so now let us see what are the divine testimonies concerning the
  grace of God, without which we are not able to do any good thing.