Please note: I approach all Scripture typologically. I also approach it as literal when the narrative calls for it, as it often does, as it definitely does here in Genesis.
Adam is a representation of Christ, so Eve is a representation of the bride of Christ, of the church. What lessons can we derive from this?
First, Eve was not created ex nihilo, but rather built out of Adam. Her existence is therefore derived from him and dependent upon him. This is a fairly obvious lesson to apply to the church; only Pelagians (who are heretics anyway: go Augustine!) would disagree.
Secondly, and somewhat speculatively, Adam was fully created before Eve was built from him. The argument could easily be made and defended (though not easily defended from this section of verses) that Eve is created in the image of Adam, and therefore in the image of God. We know that she is created in the image of God, but how dependent this is upon her husband being so created, we are not told.
Allow me to explain a bit more: had Adam been created with one leg, would the same have been true of Eve? She was, after all, taken from him, bone of his bone and flesh of his flesh, was she not? Would she not therefore reflect him?
The relevance of this is as follows: we are called to imitate Christ. We will inescapably imitate our perception of Him. We will be built in the image of our Adam. So, are we softening the gospel to make it more palatable? Then we have a skewed view of Christ. Are we harsh in defending orthodoxy? We have a skewed view of Christ. Whatever our view of Christ, we will imitate it. Practical application? Read the gospels repeatedly.
This also has a rather uncomfortable application to husbands. You are the image of Christ that your wife is being built into: her flaws reflect yours.
If this is not painful, you either have a sharp learning curve ahead or a wife that should most definitely not have settled for you (maybe she didn't get out much--that was the case with mine). Look at the mistakes that your wife has made (or is making), and TAKE RESPONSIBILITY. They are not necessarily your fault, though that is not a far-fetched explanation, but they are necessarily your responsibility--Christ died for our sins, and we are called to imitate Him. As the head, your entire family will reflect your flaws with brutal clarity. You are the spring; do not complain about the dirty water.
The final application that I will make is the one that has nearly become trite by its many trite repetitions: Eve was taken from Adam's rib so that she would be an equal--not from the foot or the head, but the side. But let us return to our original application: Eve is the Church. We are called to be an equal of Christ.
This is the future of the church, and this needs a whole lot more sermons than it has been given.
To conclude, just note the parallels between the creation of Eve and the creation of the Church:
1. Deep sleep / Death.
2. Opened side / opened side.
3. Presented to Adam by God / will be presented faultless on the last day.
From these we can see that we are in the state of our ongoing creation.
Let us then look to Christ and search out His Nature, and apply it to our lives. Then let us TEACH IT TO OUR CHILDREN. This is always the order: Learn, apply, teach. May God give us the grace to become a fitting bride for His Son.