I think you mean "circumference" not "diameter". The diameter goes through the earth and comes out on the other side, the circumference hugs the exterior of the earth and comes all the way around.
1.: Yes, they would touch, not only appear to touch, if you built the plank by having it hug the earth. If you built it by precisely measuring the angle between the planks that you are attaching to each other so that they are always exactly 0°, then your plank would eventually stop touching the earth and grow out into space and never reach the other end.
2.: Depending on the way you proceeded in building your plank (see above) the plank would either seem to be a circle or a straight tangent touching the earth at the point you started building the plank.
(Of course, wood couldn't stand the strain of its own weight after only a couple of hundred yards if you were to choose the 0° construction strategy - but I am assuming your "wood" is infinitely sturdy)