What's the difference in all the ways to say cool down?

冷める、冷ます

These 2 are used for cooling something hot, like hot water, hot food, hot engine to normal temperature.

冷める

This is intransitive verb which does not need objectives. Like "Food is cooling".

冷ます

This is transitive verb which needs objectives. Like "He's cooling the engine".

冷える、冷やす

These 2 are used for cooling something at normal temperature to cold level (like below 0 C).

冷える

This is intransitive verb again. Like "Ice cream is cooling in a freezer".

冷やす

This is transitive verb again. Like "He's cooling watermelon with ice".

冷やかす

This is not physical cooling, Rather this is cooling people's excitement. Or it's used as "letting someone down".

this is the way they translate:Cold, cool, jeer, cool, cool

冷える、冷やす、冷やかす、冷める、冷ます

i'm only a ninesei in high school but i've never seen these before. i don't think there is a way to say cool down

1) 冷える Hieru

come to be cool down

今日は冷える。

　　Kyo wa hieru.

It is cool today.

2) 冷やす Hiyasu

make something cool down　

あなたは氷で頭を冷やす必要がある。

　　Anata wa kori de atama wo hiyasu hituyo ga aru

You need to make cool your head by ice.

3) 冷やかす　Hiyakasu

　 tease somebody with slight joking

妹は私の片思いを冷かす

Imoto wa watashi no kataomoi wo hiyakasu

My sister tease me my one way love.

4) 冷める　Sameru

same as 1)

or be to feel cool

　お湯はいつか冷める

　　　Oyu wa ituka sameru

Hot water will turn to be water someday.

5) 冷ます　Samasu

almost same as 2) and

not technically but naturally

それを室温で少し冷ますことにしましょう。

　　　Sore wo situon de sukosi samasu koto ni simasho

　Let it be cool down awhile by room temperature.