While sometimes surfactants are grouped into synthetic/petrochemical and natural/oleochemical this is not a very good distinction. Natural (oleochemical) surfactants consist of a linear alkyl chains with even numbers of carbons while petrochemical surfactants can consist of linear or branched alkyl chains of even or odd numbers of carbons. However, anything made in a lab should be considered a synthetic surfactant, even if they have even numbered linear alkyl chains. However, this point doesnt seem to have any bearing on your actual question.

As I went into the way surfactants work in the last question, I will not repeat it again. Rather, I will focus on the idea that a surfactant needs to have a hydrophilic functionality as well as a hydrophobic group in order to reduce surface tension. There are polymer surfactants which contain both of these functionalities and do reduce surface tension. One example is sodium polyolefin sulfonate. In this case the polymer has an acid group making it hydrophilic while still containing a hydrophobic hydrocarbon portion. Once a polymer solidifies (crystallizes) the orientation of the polymer is set but the way in which it crystallizes can be solvent dependent. I would imagine that when a surfactant polymer solidifies it would orient itself so that the surface mainly contained either the hydrophilic or hydrophobic ends, and not a mixture of both. Further, I do not think that solid particles can be considered surfactants, even if they somehow reduce surface tension.

Using this principle, scientists can attach molecules to surfaces that either like or dislike water to varying degrees to change the surface tension of water sitting on that surface (but not the surface tension of the water with air). (Teflon (a polymer) is a good example of a coating that hates to be around anything other than itself and cause food to slip off the surface.)

Let me know if that is what you mean by solid that has surface tension reducing properties or if you would like specific examples. If you tell me what this would be used for I can help pick good molecules to fit the job.

I have attached a pdf of some pictures to help show what I mean.

For part 2, the more flat the water drop is the more it likes to be next to that surface, the more balled up it is the more it doesn't like the surface.

Making something solid would tend to jam up the fluidity of whatever it's on, which would effectively INCREASE surface tension. I am thinking for example of the difference between milk and cheese: in cheese, the bacteria have disaggregated bundles of protein fibers that are normally bound together in milk, allowing them to spread out and form a network, which increases the tension internal and external of the milk to make it a solid. There is still the same amount of water in it, though.