Are elders needed to have an "ordered" church? Are elders needed in a local church in order for it's structure to be "ordered"?

It seems that Paul was implicitly telling Titus that his work in the churches wasn't finished until elders are appointed within each congregation.


  
  Titus 1:5
  
  NASB For this reason I left you in Crete, that you would set in order what remains and appoint elders in every city as I directed you
  NIV The reason I left you in Crete was that you might put in order what was left unfinished and appoint elders in every town, as I directed you.
  ESV This is why I left you in Crete, so that you might put what remained into order, and appoint elders in every town as I directed you—
  
  


The implications of this are fairly big--saying there are a lot of "unordered" churches around: Elder-less churches, independent bible studies, etc...
I find your question as stated quite perplexing in view of the references that you give. Your question asks if the elders are necessary for and ordered church, while your references all distinguished them as two separate actions.

In order to better understand your question I have done some research into what constitutes an orderly church and what are the duties of the elders. Below are some examples of the duties of the elders as it pertains to the functions of the church.

All Scripture is quoted from the King James translation.

I have begun with an expansion of your original reference:


  Titus 1:5-9 For this cause left I thee in Crete, that thou shouldest set in order the things that are wanting, and ordain elders in every city, as I had appointed thee:  If any be blameless, the husband of one wife, having faithful children not accused of riot or unruly. For a bishop must be blameless, as the steward of God; not selfwilled, not soon angry, not given to wine, no striker, not given to filthy lucre; But a lover of hospitality, a lover of good men, sober, just, holy, temperate;  Holding fast the faithful word as he hath been taught, that he may be able by sound doctrine both to exhort and to convince the gainsayers.


After reading verses 5 through 9 I am left with a sense that Titus might have been somewhat remiss in the selection of Elders and Paul was chastising him for that, thus he reiterated his qualifications for Elders. there also may be some reinforcement for that in verse 3.


  Titus 1:3  But hath in due times manifested his word through preaching, which is committed unto me according to the commandment of God our Saviour; 


To my mind Paul might be telling Titus, that God had made Paul the church starter for the Gentiles, and that Titus was undercutting Paul's work by not appointing Church leadership in those critical areas of keeping the correct teachings in the Church.

In 1st Peter verses 1 through 9 we learn that Dependable Eldership in a church is indispensable. And Peter gives us some of the reasons for their allegiance along with the rewards they will receive.


  The elders which are among you I exhort, who am also an elder, and a witness of the sufferings of Christ, and also a partaker of the glory that shall be revealed:Feed the flock of God which is among you, taking the oversight thereof, not by constraint, but willingly; not for filthy lucre, but of a ready mind; Neither as being lords over God's heritage, but being ensamples to the flock. And when the chief Shepherd shall appear, ye shall receive a crown of glory that fadeth not away. Likewise, ye younger, submit yourselves unto the elder. Yea, all of you be subject one to another, and be clothed with humility: for God resisteth the proud, and giveth grace to the humble. Humble yourselves therefore under the mighty hand of God, that he may exalt you in due time: Casting all your care upon him; for he careth for you. Be sober, be vigilant; because your adversary the devil, as a roaring lion, walketh about, seeking whom he may devour: Whom resist stedfast in the faith, knowing that the same afflictions are accomplished in your brethren that are in the world.


We must remember that Christianity was in it's infancy and as with any new concept there were some pretty far out ideas being introduced. Christianity was subject to being completely dismantled, and dismissed as Heretical in most of the world at that time. Christianity remained in a state of turmoil up until the time of Constantine and the Council of Trent when some uniformity of belief was established. This is born out by many Church history sites on the internet, which you can google or by Church historians in innumerable books.