Our study group is trying to decipher the sources for the Gospels and wanting to know the order in which the Gospels were composed, who the proposed audiences and major themes as well.
I will answer your questions in the order that you presented them.
1. "Our study group is trying to decipher the sources for the Gospels and wanting to know the order in which the Gospels were composed..."
The widely-held critical dating of the synoptic gospels after the Jewish war 66-70AD has been described as 'an amazing example of critical dogmatism' and has been challenged by other authors, but yet held by the majority. Some argue, usually those who accept the salvation predictive element of Matthew, for earlier dating around 50-65AD. Luke is believed to be written between 75 AD and 85 AD, while others suggest an earlier date, mainly 60-61 AD for Luke's gospel. John's gospel is written last at about 95 AD and believed to be one of his last writings. Depedning what view one holds, of whether Luke and Matthew drew on Mark's gospel as a source, Mark will either be dated prior to the two or after. There is supporting evidence for both views. It is controversial, and is explained more fully with each gospel below.
Supporting data available are very slight, though. It might be argued that certain strands of Matthew tradition are of a secondary nature, in which case an interval of time would be required to allow for such developments, but this is a judgment that lacks positive proof and is controlled largely by certain presuppositions regarding the relative values of the sources used by evangelists.
Similar to this line of argument is the view that Matthew's special material shows ecclesiastical and explanatory interests' which point to a time beyond the primitive period. But again the force of this depends on the interpretation of, and the value attached to, the passages about the church. If it assumed that our Lord did not foresee and could not have predicted the mergence of a church, there would be force in the argument. But the character of Jesus would lead us to expect not only that he foresaw the future church but also even prepared for it.
Synoptic investigators so generally deny this predictive power of Jesus that it is no wonder that the dates of Mark, Matthew, and Luke are all bound together in the dating of Mark. The argument runs as follows: First, since the predictive power of Christ is denied, it is assumed that Mark was produced only a few years before the fall of Jerusalem (Mk. 13:14; Mt. 24:15). This argument is strengthened by an appeal to Matthew 2:7, which is regarded as claimed to be a reference to the fall of Jerusalem. Secondly, Matthew used Mark and therefore must be dated after the fall of Jerusalem. Thirdly, both Ignatius and the Didache appear to have cited Matthew's gospel and so the latter must have attained authority some time before the writings of the former. Fourthly, therefore the probable date of Matthew is AD 80-100. There is no general agreement on any more precise dating within this period.
Guthrie (1990) argues that if Matthew did not rely on Mark as supposed, this would call into question whether Matthew's community is closely related to post-AD 70 Judaism. It is worth raising the question whether this explanation is not necessitated by the prior rejection of an early date, rather than a supporting evidence for it.
Others argue (Guthrie, 1990) that, in fact, it is not clear that Mark's gospel must be dated just before the fall of Jerusalem in 70 AD. And the fact that Matthew used Mark is still under some dispute. Indeed, the widely-held critical dating of the synoptic gospels after the Jewish war 66-70AD has been described as 'an amazing example of critical dogmatism'. From this view, another point worth noting is that in both Matthew's and Mark's eschatological discourse there is a warning to disciples to flee to the mountains to Pella in 66 AD and this is not in the mountains, why was Matthew's statement allowed to stand if he wrote after the fall of Jerusalem. There seems no adequate answer for this.
If Matthew did not use Mark and if the predictive element is allowed there is not clear evidence to enable us to date Matthew. If Matthew was known to Luke and the latter is dated about 63 AD, this would mean that Matthew must be dated earlier than that. It has been suggested that the historic situation between 50-64 is relevant background to Matthew's gospel, in which case a date within this period would be quite reasonable (Guthrie, 1990).
It is impossible to be certain, but this should urge caution about building too much on dating. Those who place emphasis on a date for Matthew about 85AD should recognize that it is possible that that conjecture is wrong (Guthrie, 1990)
2. "...who the proposed audiences and major themes as well.."
Let's begin, then with Matthew.
A. Authorship and Date
There are mixed reviews for both the author and the date of the Book of Matthew. Traditionally, Matthew is believed to be the author of the book, although there is no evidence within the book saying who the author actually is. Saying this, though, it may be said that there is no conclusive reason for rejecting the strong external testimony regarding the authorship of Matthew, although some difficulties arise from source hypotheses. Most scholars, however, reject apostolic authorship. Yet if Matthew is not the author his identify is unknown.
The exact date is not known as mentioned above. Some argue for an earlier date 50-65AD while others for a later post-Jerusalem fall, somewhere around 85AD.
In reference to the New Testament Gospels, this solution comprehensively discusses the sources of the Gospels and the order in which the Gospels were composed, the proposed audiences and major themes as well. Supplemented with two infromative resoruces on the Gospels, which expands on this discussion.