For your first question you can give Mathematica enough information for it to be able to Solve for the coordinates of the end point that you want. Try this
Solve[{y==m*x+b,1==m*-5+b,3==m*1+b,(x-1)^2+(y-3)^2==10^2},{m,b,x,y}]
First you tell Mathematica you are dealing with a line. Then tell it about each of the two known points on that line. And finally use Pythagoras to tell it the square of the distance from your end point to the unknown point. If you run that it will give you two possible solutions. That is because it doesn't know if you want that unknown point on the line, but off to the left of (1,3) or or on the line, but off to the right of (1,3).
For your second question: you can display two graphic things on top of each other using Show. One of those can be a very simple Plot that will give you the X-Y framework that you want, but without any distracting details. The other can be the line that you want to display. Try this
Show[Plot[0,{x,-10,5},PlotRange->{-1,10}], Graphics[Line[{{2,3},{-6,10}}]]]
The Plot in that displays a horizontal line that is almost invisible because it is right on top of the x axis. The PlotRange is included to tell what range of the y axis to show, because otherwise there isn't enough information for Mathematica to guess what y range you want. Next the Graphics displays your line segment.
Look up in the help system the name of every one of those functions that I used and study the examples. See if you can try to figure out what I was thinking as I was writing this for you. Trying to understand that will hopefully help you see how to use similar thinking for you next problems.
