Qualitative scoring for responses to design fluency task
To measure the frequencies of different patterns (do some patterns occur more frequently that others based on group) I see this as a chi-squared test of independence. If you are unfamiliar with the test, a quick example is here. For your situation, all participants would get the same placements of dots, and you would count how often each possible pattern is drawn. In this design you would have two variables: 1 being control/AD, and the other being "design" with levels of every design drawn (if there are low frequencies on the designs, check out Fisher's Exact Test).

As you mention, the only way you could get enough data for RM ANOVA is to add up all of the frequencies for all of the graphs. Not knowing the theoretical backing for this measure, are all of them designed to be equivalent? At the very least it seems that the 5 dot and the 10 dot should be separated...

All my 2 cents, but on to your real questions:

1) Your "control sample" would be, in essence, a normative group (you would be determining what the "normal" person's frequencies were). These samples tend to be quite large and drawn with very rigorous designs. Check out your standard errors of your measures and see if they are acceptable (if they are, then the representativeness of your population may be your large concern with a small normative sample).

2) Double dipping is almost always a big "no-no." Idiosyncrasies in your initial "control sample" would drop down to your "clinical sample," casting doubt on the internal validity of the study. If the classifications are made for/from one group it usually follows that, in comparing to a second group, the second group with almost always not meet perfectly with the classification. In regression, this concept is that of "overfitting the model." One way you can get around this is create your classifications with, say, half of the control group, and use the other half as the clinical group. This has two problems, though: you further reduce your sample size; and I'd cast questions on the validity of "making classifications" (especially with so small a sample). Perhaps look at making all possible connections their own level of a categorical variable (combining only reflections and rotations).