[Sashimister]

３１）だが丁度その時、偶然のうまい機会が来た。私が 汗をぬぐはうとして、ハンケチで額の上をふいた時に、 帽子が頭からすべり落ちた。However, just at the right time, unexpectedly there came a promising chance. As I was sweating, just when I reached up to mop my face, the hat slipped and fell from my head.

Nicely done.

３２）それは輪のやうに転がつて行つて、すぐ五六歩後 から歩いて来る、女たちの足許に止まつた。The hat went immediately rolling off like a wheel for about five or six paces, coming to a stop right at the girls’ feet.

Not that it makes much difference, I just want to say that the original's structure after the first comma isn't reflected in the translation. The way you did it may have been intentional, therefore, no problem.

It says "it stopped at the girls' feet who were walking just five or six steps behind."

３３）若い方の娘が、すぐそれを拾つてくれた。The younger sister　picked it up at once.
Good.

３４）彼女は恥ぢる様子もなく、快活に私の方へ走つて 来た。She, without an ouce of embarrassment, ran cheerfully up to me.
Good.

３５）「どうも……どうも、ありがたう。」
私はどぎまぎしながら、やつと口の中で礼を言つた。
“Th-Thank you.”
I managed to spit out my gratitude, in a complete fluster.
Good.

３６）そして急いで帽子を被り、逃げ出すやうにすたす たと歩き出した。宇宙が真赤に廻転して、どうすれば好 いか解らなかつた。
Hurriedly I put on the hat and as though about to run away, walked briskly off. With the whole world downright wheeling around me, I had no idea if I had done the right thing. (I couldn't tell if the 真赤 here meant as I've put it, or actually something like "With the universe revolving crimson, I had~" Is there a rule of thumb for this difference in meaning, or is it all just down to context?)

真赤に modifies 廻転して, so your "With the universe revolving crimson" is good.
The phrases expresses the narrator's great confusion, embarrassment and most of all, panic. 宇宙が真赤に廻転する is in no way a common phrase but it certainly captures the speaker's mental state of the moment.

どうすれば好いか解らなかつた means "I didn't know what to do" rather than how you translated it.

３７）ただ足だけが機械的に運動して、むやみに速足で 前へ進んだ。But with just my legs moving like machines, I steamed on ahead at an absurdly fast pace.

Nice.
　
３８）だがすぐ後の方から、女の呼びかけてくる声を聞 いた。
「あの、おたづね致しますが……」
However, directly from behind me I heard a female voice calling me.
“Excuse me, I have a question…”

Good.
　
３９）それは姉の方の娘であつた。彼女はたしかに、私 よりも一つ二つ年上に見え、怜悧な美しい瞳（め）をし た女であつた。
It was the elder sister. She was certainly a year or two older than I, and had a shrewd and lovely eye to her.

Great.

４０）「滝の方へ行くのは、この道で好いのでせうか？ 」
　さう言つて慣れ慣れしく微笑した。
“If one were to be heading for the waterfall, would this be the right path?” and so saying she gave a practised smile.

Couldn't have done it better myself.