The simple and direct answer to your main question of what was the name of the other "apostle" (disciple) on the Road to Emmaus, is that the bible does not tell us. He remains unnamed.
Now, as part of your back up support, you speculate that there could have been a 13th apostle. In arriving at that speculation I believe you may have made a few assumptions that I don’t believe hold up when considering the proper biblical context.
First, you assumed that the two on the road to Emmaus were apostles, however, nowhere in the text does it state that the two were apostles. Most likely they were disciples of which there were many but they were not apostles.
Second, you assume that the people speaking in Luke 24:34 are the two men from the road to Emmaus when they are reporting what happened; you assume they make the statement “The Lord is risen indeed and hath appeared to Simon!”. However, if you consider the full context, this statement was most likely made those who were with the apostles, when the two from the road to Emmaus found them.
So based on this incorrect assumption, you then assumed that verse 35 (“And they told what things were done on the way, and how He was known to them in the breaking of bread”) indicates that Simon Peter was also there at the breaking of bread on the road to Emmaus. This cannot be true. There were only two on the road to Emmaus and the bible is clear that the two came and found the 11 (which must include Simon Peter for Judas had hanged himself before this time) and then reported what happened on the road to Emmaus. Again, verse 34 is most likely spoken by the other disciples.
So, the most logical sequence of events per the context:
- The two from the road, Cleopas and unnamed disciple, seek and find the 11 apostles and other disciples that were with the apostles
- Those that were with the apostles state that it was true that Jesus was alive and had shown Himself to Simon Peter as other scriptures attest (eg, 1 Corinthians 15:5)
- Then those two disciples from the road to Emmaus tell the events they experienced.