Transversal, I think
http://www.mathopenref.com/anglestransexterior.html
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There are three lines in the figure described. Two of the lines never meet, these are the parallel lines. The third line crosses the other lines, it is the "transversal" line. If the parallel lines are really line SEGMENTS then each can be bisected (cut into two equal lengths) This is what your description states.
Absolutely. The transversal line is the line that crosses two parallel lines. Here is a good link: * http://www.mathopenref.com/transversal.html
They are parallel to each other.
since one parallel lines is perpendicular to another line, the other parallel line is perpendicular to the line as well. so the two would not be parallel, only the original two.
If two lines are parallel to the same line, then they are parallel to each other.