冷える and 冷やす: a chilled drink and cooling it
Compare a chilled drink with cooling it, and examine 冷ます and 冷める when something cools from a hot state.
Check a chilled drink
冷蔵庫に入れたお茶がよく冷えています。 says that the tea placed in the refrigerator is well chilled. 冷える, read ひえる, describes its lowered temperature.
まだ冷えていないので、もう少し待ちましょう。 suggests waiting a little longer because it is not chilled yet. Putting it in the refrigerator and reaching the desired temperature are different events.
What is cooled and how
飲み物を氷で冷やしました。 means that someone chilled drinks with ice. 冷やす, read ひやす, marks the drink with を and the cooling means with で.
夜は足が冷えやすいです。 says that the speaker's feet tend to get cold at night. 冷える can describe body parts as well as food and does not imply intentional cooling.
When hot food cools
スープが熱いので、少し冷ましてから飲みます。 means that the speaker will let the hot soup cool a little before having it. 冷ます, read さます, suits reducing heat from something hot.
料理が冷めてしまいました。 says that the food has gone cold, with a possible sense of regret. 冷める, read さめる, overlaps with 冷える, but comparing a chilled drink with food losing its warmth helps in choosing a verb.
Does スープを少し冷ましてから飲みます necessarily mean refrigerating the hot soup?
Show the answer and why
No. It specifies cooling the soup a little before having it. The sentence does not specify how it is cooled.