Why is the Earth's shadow so "blurry"?
Because of it's atmosphere. If the earth had no atmosphere, the shadow would cut the moon's face razor sharp. Light coming around the edge of the earth is refracted as through a prism thus "smearing out" the shadow edge. In general, the sharpness of a cast shadow depends on the size of the object which casts it and the wavelength of the light illuminating that object. Since the earth is very large compared to the wavelength of visible light it should be like a razor cut on the moon ... but plainly it ain't.
Why is the moon dirty red-brown when it's in the shadow ... shouldn't it be invisible?
Yes. If the earth didn't have an atmosphere the moon would disappear in it's shadow. Let me make up a word here (there probably is aword already but I don't know it) ... UBER-UMBRA ... meaning, that part of the shadow which isn't the umbra or penumbra but rather the place wherein no visible light is refracted. (Yah know? Where 'da sun don't shine.)
How big is the earth's umbra at the moon's position?
Around 4000 miles wide. You can estimate it's size pretty well by timing the shadow's coverage of the moon. Since the moon is 2000 miles in diameter, from the time the shadow first touches it to the time that it is completely covered is about 1 hour. Therefore, the moon must be going about 2000 miles per hour. Now, it takes about 3 hours to complete the entire operation from first blemish to total uncover. So that makes it about 4000 miles wide.
How are lunar eclipses predicted?
The moon's orbit is tilted relative to the earth's orbit. Therefore, there are only two times per year when that orbit passes through the earth's umbra.
Now, when it does this ... where is the moon in that orbit? To be eclipsed it must be at that position in it's orbit that is in the umbra. If it is elsewhere there will be no eclipse ... or ... if it is close to that position there will be a partial eclipse.
Now, all you have to do is calculate where the moon will be in it's orbit using it's known velocity ... and ... from observation of the tilt of the moon's orbit, what two dates per year it goes through the umbra. (Well, one date is January 20 or close to it ... so the other must be about July 20 ... get it?) Now, get out your astronomy calculator and go figure where the moon will be those dates.
"Hmmmm ... 2000 miles per hour ... 24 hours per day ... 29 days per month ... pi times 2 times 240,000 miles ... 6 months till July 20 ... hmmmmm."