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First, I want to make the Patrol Points a certain Skill.

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Does it matter what Skill I choose for this?

To some extent, the skill selection is left up to the "client programmer".
Here a "client programmer" is defined as a lite-c programmer for a game made with 3D Gamestudio.
The "client programmer" might be you.
You might select any ENTITY skill number not already assigned or used for some other purpose.
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Is there some rule where Skill22 has to be the player?

(I do not know of any such rule.) Such rules might be set by the "client programmer".
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Should I go about setting the skill in WED or SED?

The first 20 or so skills might be set in WED. However, in the case of a simple patrol AI, a skill or skills might be more likely set via script using lite-c and SED.
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Next, how should I have the enemy's check for touching a Patrol Point?

The answer to that might depend upon the definition of "Patrol Point".
If a "Patrol Point" is an path, than path instructions might be used. If it is an ENTITY, than c_scan or other instructions might be used.
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Once the enemy knows something is nearby it, how do I have it check if the thing it's touching has a certain Skill set to a certain number?

If c_scan is used with the enemy to detect a something (ENTITY) nearby, than an event might be fired, and inside that event the "you" pointer might be set to that nearby thing (ENTITY). Skills of that "you" pointer might be evaluated.