Why does this genetic pedigree show an autosomal dominant allele at one generation but not another?
Why don't the offspring 1, 2, 3, 4, and 5 in Generation III show a similar rate of expression?


Because there are only 5 individuals in that list. The entire generation III involves 8 individuals (all the offspring of one affected and one unaffected parent). The entire pedigree involves 12 offspring of an affected and unaffected parent.

When you look at III 1-5, you see 1/5 affected (0.20).

When you look at III 1-8, you see 3/8 affected (0.375).

When you look at all offspring of an affected and unaffected parent, you see 6/12 affected (0.50). 

As you increase the number of offspring you consider, the proportion affected gets closer to the expected value of 0.5. Given a very rare autosomal dominant condition (i.e., I-2 is most likely heterozygous), this is exactly what would be expected. You may have to look at fruit flies, or some other prolific organism, to see it quite so clearly. Mendel looked at tens of thousands of pea plants, if I recall.  

Generally, you shouldn't use the proportion of offspring that show a trait to rule out any particular mode of inheritance, especially if you're looking at a small number of offspring. You can state the probability of a particular child expressing a mendelian trait. That doesn't mean a small sample won't deviate from that probability.

You can read more about this in chapter 7 of Thompson &amp; Thompson Genetics in Medicine.