I have dealt with claimed contradictions in the Bible literally hundreds of times. I have done an expose on supposed contradictions in my book Reasons for Belief (available at www.ipibooks.com). If you do a search of my web site using the word contradiction you will find many dozens of answers to supposed contradictions. My overall response is that well over 95% of all supposed contradictions are very obviously not contradictions at all–that they are so easily not seen as contradictions, that they do not even really need serious consideration. I will, however, concede that there is a rather small number of proposed contradictions which need some careful attention by believers. I have written on most of these at my web site. Each of these claims must be taken one at a time, including those you ask about above.
So, let me get to your question. On Peter’s denial of Jesus, this story was part of the general knowledge of the early Christians. That Peter denied Jesus exactly three times was well known twenty years or so before Mark of Matthew or Luke wrote their accounts. So, the question is whether there is something inherently inconsistent in the various accounts.
In Matthew 26:34 Jesus tells Peter that he will deny him three times before the rooster crows. Peter, of course, denied that this could possibly happen. So, what happened in the courtyard while Jesus was on trial? Let us see. In Matthew 26:69-75 Peter does indeed deny Jesus three times, followed by the crowing of a rooster. We are not surprised that Matthew does not contradict himself! What about the other accounts? In Mark 14:30 Jesus told Peter that before the rooster crows twice, Peter would deny him three times. This provides an extremely small additional piece of information, which is that the rooster crowed twice, with the second crowing happening immediately after the third denials (although some manuscripts of Mark do not have the word “twice,” it is likely that this was in the original Mark). No contradiction so far. Matthew does not mention the double crowing, but this is obviously not a contradiction. Matthew simply does not mention the additional detail. Then, this prediction is proved true when Peter denies Jesus three times, followed by the rooster crowing. The fact that this was the second time the rooster crowed is an absolutely insignificant difference of detail offered by Mark and not by Matthew. Now let us go to Luke to see if there is any contradiction. Again, in Luke 22:34 Jesus tells Peter that he will deny him three times before a crowing of a rooster. No contradiction here. Then, in Luke 22:54-60 Peter denies Jesus three times, followed by the crowing of a rooster. No contradiction. Then there is the account in John’s gospel. In John 13:38 Jesus tells Peter that “before the rooster crows, you will deny me three times.” Then in John 18:15-27 Peter denies Jesus three times, followed by the crowing of a rooster. Where is the supposed contradiction? I see none. Did someone tell you that the stories contradict? That person is obviously wrong. Was this claim based on the tiny difference in detail in Mark that this crowing was the second crow of a rooster that morning, which clearly does NOT contradict Matthew or Luke? This claim of a contradiction is a very weak one, for sure.
If these four accounts were exactly the same, then what would be the need for four gospels? The four gospels show evidence of being separate accounts of the same events. Therefore they contain slightly different details that one person recorded but another did not. The slightly different mention in Mark that the crowing of the rooster happened twice is an additional detail from Mark, but that is all it is–additional information not mentioned by Matthew, Luke and John, but it is not a contradiction. Period.
About John 21:14, here is what it says. “This was now the third time Jesus appeared to his disciples after he was raised from the dead. Presumably, what John is saying is that this is the third time Jesus appeared to an assembled group of apostles (plus others also in attendance?). Where is the contradiction here? You mention his appearance to the Marys. This was not an appearance to the assembled apostles for sure. You also mention his appearance to two believers on the road to Emmaus. This, too, was not an appearance to the apostles. If a skeptic claims that this is a contradiction, they need to try harder to come up with a “contradiction” than this! When John wrote his gospel he knew what he was talking about. Jesus appeared to the apostles in the upper room, but Thomas was not there. Later he appeared to the same group, with Thomas present, and then he appeared to the disciples for a third time, as recorded in John 21:14. You do not need to trouble yourself with these rather obvious non-contradictions!!! You should take such claims by skeptics with a large grain of salt. It seems that in these two cases they did not even try to spin a version of the events which is not in contradiction. If someone wants to claim a contradiction, the benefit of the doubt should be given to the writers who knew what they were talking about. Again, if these two are example of supposed contradictions for Christians to be concerned about, then you do not have all that much to be concerned about. What you need to do is what I have modeled here, which is read the supposed contradictions, then read the relevant passages and ask a simple question: Is there a reading of these separate accounts which can explain the supposed contradiction? You will find that this very simple process will resolve nearly every claim of skeptics that the Bible is full of contradictions. By the way it is NOT full of contradictions. In fact, I have not found a genuine unresolvable contradiction yet.