Why did the mid-early church consider the Gospel of Matthew to be reliable? The Gospel of Matthew is now considered to be important, however the 1st century and 2nd century references to it are surprisingly confusing and sparse. 

Which authors in the 1st and 2nd century refer to it unambiguously? For each author, please note the following: 


When they wrote. 
Whether they cite the Gospel of Matthew's content.
Whether they refer to it as the Gospel of Matthew. 
Whether they seem to have considered that it was written by Matthew the apostle. 
What criteria they use for acceptance as canonical.


Examples


Tertullian seems to have written ~190AD (?), but doesn't associate the contents of what is now the Gospel of Matthew with Matthew the person, and instead associated it with Luke.
Clement (end of 1st century?) seems to quote Matthew, but does not say it was written by Matthew, or indeed give it a name or authorship of any sort.
Papias (150AD) mentions a Gospel of Matthew, but gives the impression that it was written in Hebrew and does not cite it, so doesn't leave us certain that he was talking about the same one.
Iranaeus (190AD) gives what is most similar to the modern account of the authorship of the Gospel of Matthew. (I assume that he cites it so that we can be sure that he saw what we see in the gospel of Matthew?)


So: Are the above correct? Are there other 1st and 2nd century sources that refer to and/or cite the Gospel of Matthew and its authorship? What criteria did they use for determining authorship and canonicity?
Below is my answer to the original question: &quot;What are the arguments for Matthew's canonicity that don't appeal to Matthean authorship or the authority of the church?&quot;  I have leaving it as is, because it is still valid answer to that part of the question (and good information), although not all of it makes sense as a reply to the edited question.

First, I think it is important to emphasize that no church leader or set of leaders sat down one day and decided that X was in and Y was out.  Instead it was decided over time by the Church (fellowship of all believers) through use of some sources and disuse of others. Early leaders, of course, had some influence over what entered the canon, but no single person, congregation, or group of people decided this.
We have some extant writings about what criteria were used to decide matters.  For example, Tertullian, writing in objection to Marcion (Contra Marcion, Book 4, Ch 5), says that his followers &quot;subvert [the gospel] by shameless tampering&quot; and says

Such are the summary arguments which we use, when we take up arms against heretics for the faith of the gospel, maintaining both that order of periods, which rules that a late date is the mark of forgers, and that authority of churches which lends support to the tradition of the apostles; because truth must needs precede the forgery, and proceed straight from those by whom it has been handed on.

Various factors enter into the criteria recorded by various Church fathers:

is the document internally consistent,
is it teaching orthodoxy,
was it accepted by early churches,
and Apostolic authorship is seen as key.

This can be seen, for example, in the dismissal of an elder who confessed to writing The Acts of Paul and Thecla (Tertullian, On Baptism, Ch 17), despite the work being perfectly orthodox. In short, there was no such thing as a &quot;pious fraud&quot; who was accepted by the church. If the early Church believed Matthew was not written by Matthew, it would have never been accepted. You cannot separate the two ideas.
Turning specifically to Matthew, it was the most quoted NT work during the first two centuries and was rejected by no known group, heretical or orthodox, except for Marcion (who rejected all but his own edited versions of Paul's letters and Luke). If any work can be considered canonical by the early church, it is Matthew.  It was not accepted because of the testimony of Papias (and just because he is the first extant witness of Matthean authorship, does not mean he was the first), but rather was universally viewed as coming from Matthew the apostle.
I would strongly caution against rejecting a work because of &quot;difficulties&quot;.  Of course if you remove any gospel, the number of apparent contradictions will decrease.  However, minor contradictions show the reliability of text, not the opposite.  Real eyewitness testimony is marked by such differences; faked or non-independent accounts are not.  I would also be very cautious about accepting pseudographical claims lightly.  The case lies on a few far from certain facts and stand contrary to every source in antiquity.  I cannot go into detail here (ask another question if you like), but suffice it to say that the burden of proof should lie with those claiming pseudography (due to the frequent and early testimony to the contrary and the demonstrated disdain for the practice) and the case has not been proven.
In summary, to answer your question: besides apostolic authorship and church authority, canonicity can be argued on early and frequent use, the acceptance by early believers, internal consistency, and the orthodoxy of Matthew. However, in the views of the early Church, apostolic authorship is also essential.