Although it rained yesterday, the rain was smalllight and lasted shortly, so the coolness after the rain didn’t last long. Although it rained yesterday, the rain was light and lasted shortly, so the coolness after the rain didn’t last long.

Instead of lasted shortly I would say "the rain was light and did not last long."

Although it rained yesterday, the rain was smallhort and lasted shortlyight, so the coolness after the rain didn’t last long. Although it rained yesterday, the rain was short and light, so the coolness after the rain didn’t last long.

BeingI was under the sun for only severala few minutes, and the top of my head felt pricklingy and the skin on my face felt dry and tense. I was under the sun for only a few minutes, and the top of my head felt prickly and the skin on my face felt dry and tense.

Being underAfter being in the sun for only severala few minutes, the top of my head felt pricklingy and the skin on my face felt dry and tense. After being in the sun for only a few minutes, the top of my head felt prickly and the skin on my face felt dry and tense.

The moment I returned to house, I washed my face immediately and felt much better. The moment I returned to house, I washed my face immediately and felt much better.

Nothing is wrong but it's a bit repetitive. Maybe say it like this instead: "When I returned home, I immediately washed my face and felt much better."

In the evening, the temperature wasn’t much lower much, but the humidity increased a lot, so the air was stuffy. In the evening, the temperature wasn’t much lower, but the humidity increased a lot, so the air was stuffy.

In the evening, the temperature wasn’t much lower much, but the humidity had increased a lot, so the air was stuffy. In the evening, the temperature wasn’t much lower, but the humidity had increased a lot, so the air was stuffy.

I felt my body was sticky due to sweating. I felt my body was sticky due to sweating.

Instead "from sweating"

I felt mMy body wasfelt sticky due to sweating. My body felt sticky due to sweat.

I don't want to even move mylift a finger, not to mention cooking dinner. I don't want to lift a finger, not to mention cooking dinner.

Lifting a finger or lift a finger is an English idiom you can use here

I doidn't want to even move my finger, not to mention cooking dinner. I didn't want to even move my finger, not to mention cook dinner.

Stay in the past tense.