Are the chemical compounds in animals and plants more likely to have medicinal value than what chance would predict for any arbitrary chemical compound?

I have no hard data, but it is likely that biological chemical compounds are more useful as medicines than arbitrary non biological chemicals. Non biological chemicals would be more likely to be poisonous to all life or have other strongly negative effects with few positive effects since evolution has had no opportunity to deal with the new and novel chemicals.  Whereas biological chemicals are more likely to be compatible with life and have less intense negative effects and may have more positive effects since life has had many opportunities to encounter these kinds of chemicals and to deal with them. It is very easy to poison all living things with a nonbiological chemical, what is harder and more subtle is to find a chemical that harms one kind of living thing, the parasite, while having no effect on the host. These are the kinds of chemicals that evolution develops as defensive mechanisms all the time and therefore will be more likely to find medicinal applications.

plants provide more medicinal sources of chemical compounds than animals -- i would guess more than 80% -- unless you are asking if we extract medicinal compounds from animals. byetta, for example, is a diabetes drug extracted from the gila monster, a lizard in the southwest. but that is molecular medicine -- that is distinct from creating actual compounds that it secretes into the environment or into its extremities in response to what is out there.

Yes. They were "designed" to have some biological activity. But, they weren't meant for humans in most cases. That's why many drugs derived from natural products have to be modified to reduce off-target effects, increase potency (so the patient doesn't have to eat a huge pill), reduce toxicity, increase absorption (most people prefer oral pills, not injections), etc. There is a medicinal chemistry term for this: ADMET: absorption, distribution (within the body), metabolism, excretion (the drug should not accumulate), and toxicity.