Do you like looking at the moon？?

Use single-byte punctuation—the extra space around the double-byte question mark looks “wrong” in English.

I think about nothing in particular while I look (at it), but that peaceful time in the moonlight makes my mind calm.

Then, I can go to sleep welland sleep well/peacefully.

By the way, doeis NASA's project which theinvolves a human being standsing on the moon progressing?

Do yYou lLike lLooking at the mMoon？

All major words are capitalized in a title

I think about nothing in particular while I look, but that peaceful time in the moonlight makes my mind calm.

Then, I can go toAfterwords, I sleep well.

"Then" isn't wrong per se, but "afterwords" would fit the context of your text much better

By the way, doeis NASA's project which the human being stands on the moon progressmoon landing project still ongoing?

"By the way" is used when speaking to someone, but the rest of your text feels like an internal dialogue and almost poetic. I'd rather use something like "I wonder if NASA's moon landing project is still ongoing" to continue with that feeling.