I must wash it and dry it in the sun, but the sun is not coming out for a while according to today's weather reports.

I think this would be more natural rephrased like, "I must wash it and dry it in the sun, but according to today's weather report, the sun will not come out for a while." That way, the main idea of the sentence (the sun will not come out for a while) is emphasized because it's at the end of the sentence.

The mild and conmformtable weather won't be here until autumn comes.

It has been cloudy lately, and I am feeling quite depressed.

I agree with thomas256

There is already a lot of humidity in our house, and the bamboo steamer grew mold.

I agree with thomas256, it is probably more common to say the bamboo steamer "got moldy". You could also say "the bamboo steamer started growing mold".

I must wash it and dry it in the sun, but the sun is notwon't be coming out for a while according to today's weather reports.

I agree with thomas256, and would add that "the sun won't be coming out for a while" sounds more natural

However, once the rainy season ends, the scorching heat of summer arrives, making it impossible to walk around in the street.

The mild and conformable weather won't be here until autumn comes.

I think it would be more natural to say "We won't get mild and comfortable weather until autumn comes."

Also, with energy prices rising, air-conditioning costs canmay be head-wrenching duringworrisome this summer.

I agree with thomas256, "head-wrenching" is not an actual phrase. Did you mean 頭が痛い? You could use 'worrisome' or troubling' perhaps?

I truly feel that we humans have to face the harshness of nature.

I agree with thomas256's comment here. Alternatively, may you could say "I truly feel that we humans are forced to face the harshness of nature."?

It has been cloudy lately, and I a'm feeling quite depressed.

I might say "so" instead of "and" if these two concepts follow a cause-and-effect relationship. "It has been cloudy lately, so I'm feeling quite depressed."

Also, it's far more natural to use contractions like "I'm".

There i's already a lot of humidity in our house and the bamboo steamer grewot moldy.

We rarely say that something "grows mold". Instead, things "get moldy".

I musthave to wash it and dry it in the sun, but the sun is non't coming out for a while according to today's weather reports.

"I must" sounds kind of dramatic and poetic. Like you need to do this thing, and you're GOING to do it no matter WHAT happens. It carries a solemn sense of duty. "I have to" is more common.

Once the weather clears up, I wi'll be busy doing laundry and walking outside.

However, once the rainy season ends, the scorching heat of summer will arrives, making it impossible to walk around in the street.

The mild and conmformtable weather won't be here until autumn comes.

Also, with energy prices rising, air-conditioning costs can be head-wrenchache inducing during thise summer.

"Head-wrenching" is not an actual phrase, although I'm not exactly sure what to replace it with.

I truly feel that we humans have to face the harshness of nature.

This sentence is grammatically correct, but feels just a little off. Maybe something like: "I truly feel that we humans SHOULD JUST face the harshness of nature"?