What are the meanings of various reference lines and sections in a take-off distance graph?
You need to read such graphs strictly from left to right. Each new panel adds the influence of another parameter, and when you are done, the result is corrected for altitude, take-off weight, wind and obstacle height. Except for the leftmost panel, the groups of lines only help you to find the correct slope and do not represent a certain altitude.

You start with the outside air temperature. Pick what the instrument in your cockpit indicates, and go from the temperature axis straight up until you intercept the line which is closest to your given pressure altitude. This can be read directly from the altimeter if it is set correctly. You can try to interpolate between lines, but to be safe pick the next higher value.

Next, move horizontally to the first reference line. From there, follow the slope of the lines until you intercept the x-position which corresponds to your aircraft's mass. If you start between two lines, you should move such that the relative vertical distance to the neighboring lines does not change. Note that the take-off mass is also a value you should know already if your preflight was done properly.

From that point, jump off horizontally until you intercept the next reference line. Depending on wind direction, follow either the dashed or the solid lines until you arrive at the x-position which corresponds to the wind component in runway direction. You will note that for headwind you will move slightly down, so headwind shortens the take-off distance. Makes sense, right?

The last panel helps you to find both the take-off roll (obstacle height zero) and the take-off distance over a 50 ft obstacle. Here you can only extract the values at the x-positions 0 and 50 and interpolation will not work. The y-axis all the way on the right now will tell you the distances in feet.

The reference lines show the values for the standard take-off at MTOW and calm wind. The examples given by the red lines are for two different situations:


Solid red line: OAT is 90°F, pressure altitude is 2000ft MSL, take-off mass is 2500lbs and with a headwind of 20kts the take-off roll is 500ft. At this point your speed should be 62kts (see table).  This is the result for your first question.
Dashed red line: OAT is 15°C, pressure altitude is 6000ft MSL (rounded up from 5650ft), take-off mass is 2950lbs and with a headwind of 9kts your take-off roll is 1400ft and the distance needed to clear the 50ft obstacle at 1.3*stall speed is 2300ft.