How could you find the distance to the moon, knowing only its diameter, and having only a meter stick and pen?

it is not possible with just a meter stick and a pen, but is possible if you have a theodelite or a protractor.
measure the angular span of the moon using a theodelite. Let that value be x degrees. let the diameter of the moon be 'a' km, and its distance 'b' km.
now,
tan x = a/b
so,
b = a/tan x

(tan is a trigonometric value, which can be calculated using a calculator.)

Yes.  If you hold the ruler at arm's length, there are two similar isosceles triangles--from your eye to the ruler, and from your eye to the diameter of the moon.  If we call:

r = distance from hand to eye
R = distance from moon to eye
d = diameter of moon seen on ruler
D = real diameter of moon.

You can imagine then the two triangles with the same angle at your eye with legs r and base d, and with legs R and base D.  That means:

R / r = D / d since the two triangles are similar

Therefore:

R = r * D / d

Since the moon is about 2200 miles in diameter, and it is typically a finger's-width at arm's length, or 0.5 inch at 3 feet, this means

R = 36in * 2200 miles / 0.5 in = 158,400 miles

which is not really anywhere near the real value of 230,000 miles, but I'm only guessing at these numbers because it's daytime