First, about the "rejected gospels", at the time when the bible was put together there were a few gospels around of which many were written long after the eye wittnesses were all dead which doesn't make them very trustworthy. So what they did was to choose four gospels, that were all written when eye witnesses were still around, and have these as standards cause they were the most trustworthy. So really no "choosing of the ones fitting the catholic church the best". More like neglecting the ones written over hundreds of years after Jesus' death and resurrection. So choosing the most trustworthy.
You make a good point here. Many other gospels were rejected because they couldn't be trusted, but then the 4 gospels we know today were also not that believable as a source of facts. As always, anyone can find faith in them, if that's what you seek.
The gospel according to Mark, which supposedly is the earliest, was written (guessed to be written) around 70AD. This was followed by the gospels of Matthew and Luke, written probably sometime between 70AD and 100AD. And John's was probably the last one written, at the end of the century.
There is a high possibility that Mark's, Matthew's and Luke's gospels share a common source, or that the later two, were based partialy on Mark. The reasoning being that there are just too many similar passages. Either way, most people during the time couldn't write, so until the stories were written down, they were most likely passed by word of mouth. And that likely made the stories undergo small, but frequent alterations, until it was probably hard to tell what had been added, and what had not.
Also, if you think of it, the writers of these gospels were early christians, at the time that christianity was expanding. The whole point of the gospels was to inspire people to believe in Jezus and to join the faith. The tools of almost all religious writers at the time was to include symbolic elements, such as magical numbers, places, dates etc. The romans, whom the gospels for a large part were for, as that's the birth place of early christianity, were used to these symbolisms, from their own beliefs, and it's logical that when trying to teach a faith, you use the right words for your listeners.
Another last interesting fact: The 4 canonical gospels were originally untitled and anonymous until 180AD when suddenly they were given the names that we now know them by. So it's possible that some of the gospels may have been written by Jezus's actual disciples, but perhaps that too, is just to inspire faith.
The four gospels were actually used very early on by the early christians! What about them makes them not as believable as a source of fact? And just as with the rest of the bible they are not written as a collection of facts but as testimonies.
The writer of the gospel of Mark is according to the tradition the assistant of Peter (since he probably couldn't write as he was only a fisherman), and Mark himself is mentioned in at least one (don't know how many) of Pauls letters in the new testament. But writing the gospels were probably sort of a "plan B" since most of the early christian leaders were absolutely sure that Christ was to return during their lifetime - eg Paul.
The "small frequent alterations" are things I've never heard of but it's easy to imagine that such a thing could happen. Though most of the stories were passed on by the disciples while teaching and preaching and since a lot of people that were around Jesus still were around when the gospels were brought to use any big and vivid "alterations" would probably have been corrected by one of the christian leaders e g Paul who frequently wrote letters to the different christian parishes.
About the dates - they are very disputed and some things point on the fact that they are written earlier than you say. Which only makes it more likely that disciples and eyewitnesses were still around.
About how the gospels were written - can all to well have been affected by where where they were written and to whom it was ment to reach. E g I heard that the gospel of John was written to for the greeks - and that's just logical but has more to do with what style of writing you use and not the content of what you write (eg the beginning of the gospel of John talks of the word - logogs - a very important thing to the filosofical greeks). You don't write a children's book with buraucratic text and you don't write an instruction manual with children's lanugage - but you don't change the actual content.
But one must also take into concideration that the christians were persecuted by the romans, and trying to spread the gospel was a thing you did with a high risk for your life.
And about the names - I've never heard that they didn't get their names until then. I have heard though of earlier accounts of the names of the gospels so I guess that's disputable. But it's not that long a time though ca 100 years. And I guess the names of the authors weren't the most important thing, but at least there must have been a traditional knowledge of who wrote which!