Applying for Schengen Visa while near I-20 expiration. Would it affect my application?

Yes, since it seems that a valid I-20 is needed to re-enter the United States, it forms a part of the travel document that the Schengen Code requires to be valid 3 months after the expected exit from the Schengen Area.

So if your travel itinerary includes a return back to the United States, then the conditions of Article 6 (1)(a)(i) of the Schengen Code must be fulfilled.

The wording of your quote (which I found at a Spanish Consulate site) is slightly different than that of the Visa requirements pdf of the German Missions in the United States, where plus endorsed I-20 is stated and no meantion that the F-1 must be valid for a further 3 months.

The Schengen Code does not deal with the details of what is considered a valid foreign residence permit for reentry into that country.

The Schengen Code does require that the persons travel document must be valid to return after that person's visit plus 3 months (Article 6 (1)(a)(i)).

Based on the Wikipedia I-20 (form) article, a valid I-20 is needed to reenter the United States. So the I-20 will be considered part of the travel document.

Schengen VISA (all stays of up to 90 days)
...

• Original of valid US alien registration card or valid US resident visa (type A, E, F plus endorsed I-20, G, H, I, J plus J1 document, L, O, R) and one copy

Travel and travel signatures
Whenever the student re-enters the United States after traveling, the student must have all of these at the time of arrival at the port of entry:

• A valid passport (valid for at least six more months)
• A valid F or M visa
• A valid I-20 (i.e., an I-20 whose program end date has not yet arrived)
• A travel signature on the I-20 (page 3) from the DSO that is at most one year old (six months in the case of students currently on post-completion Optional Practical Training)

...

Sources:

• PDF Schengen VISA - German Missions in the United States
• Spain - Boston Visas
• I-20 (form)