Hi A.S.,
I echo what Nigel said as being most important. However, I'll also give a little extra input regarding your specific questions:
1.) How important is church membership?
Well, in the grand scheme of eternity, it's not at all important. This is because your "membership" isn't based upon things like attendance, offerings, doctrinal adherence, etc. It's between you and the Lord.
In a practical way, many churches have "membership" rolls so that individuals who fellowship there and adhere to the beliefs and practices of that church can vote on matters of importance. If it was open voting for anyone and everyone, then unbelievers could come into your midst and vote out pastors and make changes that are, well, terrible. So, in this sense, they see an official church membership roll as a means to provide this voice to specific individuals.
It's definitely a question that a lot of people think about. In many denominations or even local congregations, the people who are offered the opportunity to work in ministry must abide by the beliefs of that congregation or denomination. You might be permitted to be involved; however, you might not in a more formal manner.
2.) If we're all in one body of Jesus, is having my name registered at a local church really that important?
Spiritually speaking, it's not important. The only roll that matters is the Book of Life. I will say that I knew a missionary who struggled with this. He was very involved in churches of a particular church organization. He went to Mexico and funded the construction of many, many church buildings (and other buildings and construction projects). He was not a licensed or ordained minister in the denomination that he fellowshipped with.
One day, this missionary received a call from the denomination's (or, as they prefer, "fellowship's") headquarters. They had heard many great things about him. They wanted to fully fund his way through Bible college and then help him receive his ordination and license (which does have some usefulness when operating overseas on certain visas). They were going to help him out (he was married and had three young children).
This missionary prayed about this. He also considered the cost and benefits. Ultimately, he chose NOT to accept this offer. He told me that he felt that it would only hinder the work that he was already doing. It would take years to finish Bible college. The denomination had a set of requirements for missionaries too -- which would take years.
So, this missionary continued to work as he had. He continued to personally raise the funds for building projects, organize teams (including for preaching, construction, medical needs and even general work). The interesting thing is that nearly all of the churches that he built in Mexico and elsewhere in Latin America were given to local congregations within that particular denomination/fellowship.
So, there is that consideration.
3.) Where in the bible is there a strong model for membership at a local church?
I think that you're right. I've read Acts many times. There was "decency and order." There were apostles, prophets, evangelists, pastors and teachers. There were individuals with different gifts too. There are very specific requirements for ministry mentioned by Paul and in other epistles. However, I don't know with any certainty that the book of Acts (or Epistles) sets up a hierarchy beyond what we read.
4.) If I informed the pastor I'm attending elsewhere, is there a need for a formal exit process?
I'm not really sure. I've heard of people who "transferred" their membership from one church to another. However, that was usually within the denomination itself. When I moved, I never really even thought about doing that. Since then, I have never actually become a "voting member" of any church that I've fellowshipped with.