I write this journal on the way to and back to myfrom work.

Same correction as previous person.

US alternative

I'm writing this journal on the way to and from work.

In Japan, Mmost trains have the placea car where only women can ride on.

or another US alternative

In Japan, most trains have a car where only women are allowed to ride.

I always choose thiso ride in one because Iit makes me feel more comfortable staying there.

From your 日本版：「いつもそこを選んで乗ることにしている」is loosely translated to "I always choose to ride there" but we can reference the womens-only car as "choose to ride in one."

I wish I could relive the experiences.

or

I wish I could remember back to those times.

From "電車に乗っていると日記で書きたいことがたくさん頭に浮かんでくる," we can say, "Lots of things come to mind for what I want to write in my journal when I ride on the train," or "When I ride the train, many things come to mind for what I want to write in my journal."

My rule for this challenge, Not to is to not use a dictionary, and depend only only my memory.

or from "この挑戦にあたっての自分のルールは辞書を使わないこと。自分の記憶だけに頼ること,"

"My rule for this challenge is to not use a dictionary. Rely only upon my memory."

I write this journal on the way to and back tofrom my work.

In Japan, Mmost trains have the place where a carriage only women can ride on.

I choose thisose ones because I feel comfortable staying there.

I wish I could re-experience those times.

Not sure what you meant here, and your original doesn't seem to mention this line, so I just tried my best to guess what you meant.

My rule for this challenge, N is to not to use a dictionary, and depend on only my memory.