My aunt and I walked for a long distance with my aunt to go shopping and to getting exercise. My aunt and I walked for a long distance to go shopping and to get exercise.

you can use either to go shopping and to get exercise or say going shopping and getting exercise. The verbs in a sentence must be of the same form

I walked long distanceent on a walk with my aunt to go shopping and getting exercise. I went on a walk with my aunt to go shopping and get exercise.

"I walked a long distance" is okay too. I just read a newer post of yours, and it seems like you walk often. So I think "went on a walk" or "went for a walk" sounds the most natural. In the same way you say "go shopping," you can say "get exercise." You can even leave out "get" and it will sound natural: "... to go shopping and exercise." When you use "get", the word "exercise" is a noun. If you don't use "get," the word "exercise" is a verb.

I walked a long distanceway with my aunt to go shopping and to getting exercise. I walked a long way with my aunt to go shopping and to get exercise.

After we came back home, I ate carry rice and cake.

Your sentence is grammatically correct , but writing I ate carry rice and cake for dinner improves the flow of the paragraph

After we came back home, I ate caurry rice and cake. After we came back home, I ate curry rice and cake.

In Japanese, "curry rice" is spelled with「カ」, which I think is where your "a" in "carry rice" came from. It's just a typo or spelling error :) In English "carry" is a different word -- a verb.

After we came back home, I ate caurry and rice , and then cake. After we came back home, I ate curry and rice, and then cake.

After dinner, my friend helped me to do English homework. After dinner, my friend helped me do English homework.

After dinner, my friend helped me to do English homework. After dinner, my friend helped me do English homework.

My English homeworkIt was watching video about coronavirus and answering questions. It was watching video about coronavirus and answering questions.

My English homework was watching a video/videos about coronavirus and answering questions. My English homework was watching a video/videos about coronavirus and answering questions.

It was difficult for me to understand the fast speaker. or It was difficult for me to understand fast speakers. It was difficult for me to understand the fast speaker. or It was difficult for me to understand fast speakers.

It was difficult for me to understand the fast speaker. It was difficult for me to understand the fast speaker.

"speaker" is a singular noun, so you need to use an article before it.

IThey/he/she spoke fast, so it was difficult for me to understand fast speaker. They/he/she spoke fast, so it was difficult for me to understand.

I thought again my English skill is low level. I thought again my English skill is low.

I thought again that my English skillevel is low level. I thought again that my English level is low.

There are many ways to say the same thing. What you said was not wrong, but it seems a little unnatural.

I thought again my English skillevel is low level. I thought again my English level is low.