What makes a question “foolish and unlearned”?
Questions that are divisive and produce bickering and arguing over words, creating a spiritually unhealthy atmosphere. Hence, Paul stressed the need to “reject foolish and ignorant debates.”
"Unlearned" is KJV Paul-speak for "ignorant," re Proto Indo-European "gno-," to "know"--a very old word which has kept its meaning for millennia. Paul is advising a believer (a Hebrew meaning of which is "to be firm," and more specifically of the OE sense of "be-love") to eschew agonism with those who have not the inner kingdom sense of "I AM"--the "still, small voice." The original group of followers of Jesus were primarily Jews, and their sect was latterly accepted/tolerated by other Jewish sects, and known as "the people of the Way." "Foolish" has as its primal meaning, in the PIE "bhel-," "to swell [e.g. with pride]," "to blow [e.g., as a 'blowhard']." Paul, likely the leading scholar of his generation (studied with Gamaliel, earning the equivalent of 2 Ph.D.s; https://en.wikipedia.org/wiki/Gamaliel ), and well-versed in e.g. the Greek philosophy and literature of Timothy's Greek father's heritage, simply advises the young "person of the Way" not to waste time re people who are not interested in receiving the Presence of Messias, "this Mind which was also in Christ Jesus." Semi-related: "The Great Divorce;" "Mere Christianity."